diff --git a/src/frontends/lean/notation_cmd.cpp b/src/frontends/lean/notation_cmd.cpp
index dfaa18bcca..b46c0a3739 100644
--- a/src/frontends/lean/notation_cmd.cpp
+++ b/src/frontends/lean/notation_cmd.cpp
@@ -27,6 +27,7 @@ static name g_infixl("infixl");
 static name g_infixr("infixr");
 static name g_postfix("postfix");
 static name g_notation("notation");
+static name g_call("call");
 
 static std::string parse_symbol(parser & p, char const * msg) {
     name n;
@@ -74,6 +75,7 @@ using notation::mk_binders_action;
 using notation::mk_exprs_action;
 using notation::mk_scoped_expr_action;
 using notation::mk_skip_action;
+using notation::mk_ext_lua_action;
 using notation::transition;
 using notation::action;
 
@@ -180,6 +182,10 @@ static action parse_action(parser & p, buffer<expr> & locals, buffer<token_entry
         if (p.curr_is_numeral()) {
             unsigned prec = parse_precedence(p, "invalid notation declaration, small numeral expected");
             return mk_expr_action(prec);
+        } else if (p.curr_is_string()) {
+            std::string fn = p.get_str_val();
+            p.next();
+            return mk_ext_lua_action(fn.c_str());
         } else if (p.curr_is_token_or_id(g_scoped)) {
             p.next();
             return mk_scoped_expr_action(mk_var(0));
@@ -218,6 +224,11 @@ static action parse_action(parser & p, buffer<expr> & locals, buffer<token_entry
                 }
                 p.check_token_next(g_rparen, "invalid scoped notation argument, ')' expected");
                 return mk_scoped_expr_action(rec, prec ? *prec : 0);
+            } else if (p.curr_is_token_or_id(g_call)) {
+                p.next();
+                name fn = p.check_id_next("invalid call notation argument, identifier expected");
+                p.check_token_next(g_rparen, "invalid call notation argument, ')' expected");
+                return mk_ext_lua_action(fn.to_string().c_str());
             } else {
                 throw parser_error("invalid notation declaration, 'foldl', 'foldr' or 'scoped' expected", p.pos());
             }
diff --git a/src/frontends/lean/parse_table.cpp b/src/frontends/lean/parse_table.cpp
index d14aee9d34..815d6f85bf 100644
--- a/src/frontends/lean/parse_table.cpp
+++ b/src/frontends/lean/parse_table.cpp
@@ -4,6 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 
 Author: Leonardo de Moura
 */
+#include <string>
 #include <utility>
 #include "util/rb_map.h"
 #include "util/sstream.h"
@@ -58,6 +59,12 @@ struct ext_action_cell : public action_cell {
         action_cell(action_kind::Ext), m_parse_fn(fn) {}
 };
 
+struct ext_lua_action_cell : public action_cell {
+    std::string m_lua_fn;
+    ext_lua_action_cell(char const * fn):
+        action_cell(action_kind::LuaExt), m_lua_fn(fn) {}
+};
+
 action::action(action_cell * ptr):m_ptr(ptr) { lean_assert(ptr); }
 action::action():action(mk_skip_action()) {}
 action::action(action const & s):m_ptr(s.m_ptr) { if (m_ptr) m_ptr->inc_ref(); }
@@ -82,6 +89,10 @@ ext_action_cell * to_ext_action(action_cell * c) {
     lean_assert(c->m_kind == action_kind::Ext);
     return static_cast<ext_action_cell*>(c);
 }
+ext_lua_action_cell * to_ext_lua_action(action_cell * c) {
+    lean_assert(c->m_kind == action_kind::LuaExt);
+    return static_cast<ext_lua_action_cell*>(c);
+}
 unsigned action::rbp() const { return to_expr_action(m_ptr)->m_rbp; }
 name const & action::get_sep() const { return to_exprs_action(m_ptr)->m_token_sep; }
 expr const & action::get_rec() const {
@@ -94,6 +105,7 @@ bool action::use_lambda_abstraction() const { return to_scoped_expr_action(m_ptr
 expr const & action::get_initial() const { return to_exprs_action(m_ptr)->m_ini; }
 bool action::is_fold_right() const { return to_exprs_action(m_ptr)->m_fold_right; }
 parse_fn const & action::get_parse_fn() const { return to_ext_action(m_ptr)->m_parse_fn; }
+std::string const & action::get_lua_fn() const { return to_ext_lua_action(m_ptr)->m_lua_fn; }
 bool action::is_compatible(action const & a) const {
     if (kind() != a.kind())
         return false;
@@ -102,6 +114,8 @@ bool action::is_compatible(action const & a) const {
         return true;
     case action_kind::Ext:
         return m_ptr == a.m_ptr;
+    case action_kind::LuaExt:
+        return get_lua_fn() == a.get_lua_fn();
     case action_kind::Expr:
         return rbp() == a.rbp();
     case action_kind::Exprs:
@@ -124,6 +138,7 @@ void action_cell::dealloc() {
     case action_kind::Exprs:      delete(to_exprs_action(this)); break;
     case action_kind::ScopedExpr: delete(to_scoped_expr_action(this)); break;
     case action_kind::Ext:        delete(to_ext_action(this)); break;
+    case action_kind::LuaExt:     delete(to_ext_lua_action(this)); break;
     default:                      delete this; break;
     }
 }
@@ -154,6 +169,7 @@ action mk_scoped_expr_action(expr const & rec, unsigned rb, bool lambda) {
     return action(new scoped_expr_action_cell(rec, rb, lambda));
 }
 action mk_ext_action(parse_fn const & fn) { return action(new ext_action_cell(fn)); }
+action mk_ext_lua_action(char const * fn) { return action(new ext_lua_action_cell(fn)); }
 
 struct parse_table::cell {
     bool                                                         m_nud;
@@ -196,7 +212,7 @@ static void validate_transitions(bool nud, unsigned num, transition const * ts,
         case action_kind::Binder: case action_kind::Binders:
             found_binder = true;
             break;
-        case action_kind::Expr: case action_kind::Exprs: case action_kind::Ext:
+        case action_kind::Expr: case action_kind::Exprs: case action_kind::Ext: case action_kind::LuaExt:
             nargs++;
             break;
         case action_kind::ScopedExpr:
@@ -296,6 +312,14 @@ static int mk_scoped_expr_action(lua_State * L) {
     bool lambda = (nargs <= 2) || lua_toboolean(L, 3);
     return push_notation_action(L, mk_scoped_expr_action(to_expr(L, 1), rbp, lambda));
 }
+static int mk_ext_lua_action(lua_State * L) {
+    char const * fn = lua_tostring(L, 1);
+    lua_getglobal(L, fn);
+    if (lua_isnil(L, -1))
+        throw exception("arg #1 is a unknown function name");
+    lua_pop(L, 1);
+    return push_notation_action(L, mk_ext_lua_action(fn));
+}
 static int is_compatible(lua_State * L) {
     return push_boolean(L, to_notation_action(L, 1).is_compatible(to_notation_action(L, 2)));
 }
@@ -329,6 +353,10 @@ static int use_lambda_abstraction(lua_State * L) {
     check_action(L, 1, { action_kind::ScopedExpr });
     return push_boolean(L, to_notation_action(L, 1).use_lambda_abstraction());
 }
+static int fn(lua_State * L) {
+    check_action(L, 1, { action_kind::LuaExt });
+    return push_string(L, to_notation_action(L, 1).get_lua_fn().c_str());
+}
 
 static const struct luaL_Reg notation_action_m[] = {
     {"__gc",                   notation_action_gc},
@@ -341,6 +369,7 @@ static const struct luaL_Reg notation_action_m[] = {
     {"initial",                safe_function<initial>},
     {"is_fold_right",          safe_function<is_fold_right>},
     {"use_lambda_abstraction", safe_function<use_lambda_abstraction>},
+    {"fn",                     safe_function<fn>},
     {0, 0}
 };
 
@@ -357,6 +386,7 @@ static void open_notation_action(lua_State * L) {
     SET_GLOBAL_FUN(mk_expr_action,         "expr_notation_action");
     SET_GLOBAL_FUN(mk_exprs_action,        "exprs_notation_action");
     SET_GLOBAL_FUN(mk_scoped_expr_action,  "scoped_expr_notation_action");
+    SET_GLOBAL_FUN(mk_ext_lua_action,      "ext_action");
 
     push_notation_action(L, mk_skip_action());
     lua_setglobal(L, "Skip");
@@ -373,6 +403,7 @@ static void open_notation_action(lua_State * L) {
     SET_ENUM("Binders",     action_kind::Binders);
     SET_ENUM("ScopedExpr",  action_kind::ScopedExpr);
     SET_ENUM("Ext",         action_kind::Ext);
+    SET_ENUM("LuaExt",      action_kind::LuaExt);
     lua_setglobal(L, "notation_action_kind");
 }
 
diff --git a/src/frontends/lean/parse_table.h b/src/frontends/lean/parse_table.h
index f2482483bb..d487a10fdb 100644
--- a/src/frontends/lean/parse_table.h
+++ b/src/frontends/lean/parse_table.h
@@ -5,6 +5,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Author: Leonardo de Moura
 */
 #pragma once
+#include <string>
 #include <utility>
 #include "util/buffer.h"
 #include "util/lua.h"
@@ -17,7 +18,7 @@ class parser;
 namespace notation {
 typedef std::function<expr(parser &, unsigned, expr const *, pos_info const &)> parse_fn;
 
-enum class action_kind { Skip, Expr, Exprs, Binder, Binders, ScopedExpr, Ext };
+enum class action_kind { Skip, Expr, Exprs, Binder, Binders, ScopedExpr, Ext, LuaExt };
 struct action_cell;
 
 /**
@@ -63,6 +64,7 @@ public:
     friend action mk_binders_action();
     friend action mk_scoped_expr_action(expr const & rec, unsigned rbp, bool lambda);
     friend action mk_ext_action(parse_fn const & fn);
+    friend action mk_ext_lua_action(char const * lua_fn);
 
     action_kind kind() const;
     unsigned rbp() const;
@@ -72,6 +74,7 @@ public:
     bool is_fold_right() const;
     bool use_lambda_abstraction() const;
     parse_fn const & get_parse_fn() const;
+    std::string const & get_lua_fn() const;
 
     bool is_compatible(action const & a) const;
 };
@@ -83,6 +86,7 @@ action mk_binder_action();
 action mk_binders_action();
 action mk_scoped_expr_action(expr const & rec, unsigned rbp = 0, bool lambda = true);
 action mk_ext_action(parse_fn const & fn);
+action mk_ext_lua_action(char const * lua_fn);
 
 class transition {
     name           m_token;
diff --git a/src/frontends/lean/parser.cpp b/src/frontends/lean/parser.cpp
index d6d9df29f8..61b6790f61 100644
--- a/src/frontends/lean/parser.cpp
+++ b/src/frontends/lean/parser.cpp
@@ -64,6 +64,42 @@ parser::no_undef_id_error_scope::~no_undef_id_error_scope() {
     m_p.m_no_undef_id_error = m_old;
 }
 
+static char g_parser_key;
+void set_global_parser(lua_State * L, parser * p) {
+    lua_pushlightuserdata(L, static_cast<void *>(&g_parser_key));
+    lua_pushlightuserdata(L, static_cast<void *>(p));
+    lua_settable(L, LUA_REGISTRYINDEX);
+}
+
+parser * get_global_parser_ptr(lua_State * L) {
+    lua_pushlightuserdata(L, static_cast<void *>(&g_parser_key));
+    lua_gettable(L, LUA_REGISTRYINDEX);
+    if (!lua_islightuserdata(L, -1))
+        return nullptr;
+    parser * p = static_cast<parser*>(const_cast<void*>(lua_topointer(L, -1)));
+    lua_pop(L, 1);
+    return p;
+}
+
+parser & get_global_parser(lua_State * L) {
+    parser * p = get_global_parser_ptr(L);
+    if (p == nullptr)
+        throw exception("there is no Lean parser on the Lua stack");
+    return *p;
+}
+
+struct scoped_set_parser {
+    lua_State * m_state;
+    parser *    m_old;
+    scoped_set_parser(lua_State * L, parser & p):m_state(L) {
+        m_old = get_global_parser_ptr(L);
+        set_global_parser(L, &p);
+    }
+    ~scoped_set_parser() {
+        set_global_parser(m_state, m_old);
+    }
+};
+
 parser::parser(environment const & env, io_state const & ios,
                std::istream & strm, char const * strm_name,
                script_state * ss, bool use_exceptions, unsigned num_threads,
@@ -694,10 +730,32 @@ expr parser::parse_notation(parse_table t, expr * left) {
             args.push_back(r);
             break;
         }
+        case notation::action_kind::LuaExt:
+            if (!m_ss)
+                throw parser_error("failed to use notation implemented in Lua, parser does not contain a Lua state", p);
+            using_script([&](lua_State * L) {
+                    scoped_set_parser scope(L, *this);
+                    lua_getglobal(L, a.get_lua_fn().c_str());
+                    if (!lua_isfunction(L, -1))
+                        throw parser_error(sstream() << "failed to use notation implemented in Lua, Lua state does not contain function '"
+                                           << a.get_lua_fn() << "'", p);
+                    lua_pushinteger(L, p.first);
+                    lua_pushinteger(L, p.second);
+                    for (unsigned i = 0; i < args.size(); i++)
+                        push_expr(L, args[i]);
+                    pcall(L, args.size() + 2, 1, 0);
+                    if (!is_expr(L, -1))
+                        throw parser_error(sstream() << "failed to use notation implemented in Lua, value returned by function '"
+                                           << a.get_lua_fn() << "' is not an expression", p);
+                    args.push_back(to_expr(L, -1));
+                    lua_pop(L, 1);
+                });
+            break;
         case notation::action_kind::Ext:
             args.push_back(a.get_parse_fn()(*this, args.size(), args.data(), p));
             break;
         }
+
         t = r->second;
     }
     list<expr> const & as = t.is_accepting();
@@ -979,7 +1037,6 @@ bool parser::parse_commands() {
     return !m_found_errors;
 }
 
-
 bool parse_commands(environment & env, io_state & ios, std::istream & in, char const * strm_name, script_state * S, bool use_exceptions,
                     unsigned num_threads) {
     parser p(env, ios, in, strm_name, S, use_exceptions, num_threads);
@@ -995,4 +1052,18 @@ bool parse_commands(environment & env, io_state & ios, char const * fname, scrip
         throw exception(sstream() << "failed to open file '" << fname << "'");
     return parse_commands(env, ios, in, fname, S, use_exceptions, num_threads);
 }
+
+static int parse_expr(lua_State * L) {
+    script_state S = to_script_state(L);
+    int nargs = lua_gettop(L);
+    expr r;
+    S.exec_unprotected([&]() {
+            r = get_global_parser(L).parse_expr(nargs == 0 ? 0 : lua_tointeger(L, 1));
+        });
+    return push_expr(L, r);
+}
+
+void open_parser(lua_State * L) {
+    SET_GLOBAL_FUN(parse_expr, "parse_expr");
+}
 }
diff --git a/src/frontends/lean/parser.h b/src/frontends/lean/parser.h
index a9ce8ffd98..10ea2a9111 100644
--- a/src/frontends/lean/parser.h
+++ b/src/frontends/lean/parser.h
@@ -270,4 +270,5 @@ bool parse_commands(environment & env, io_state & ios, std::istream & in, char c
                     script_state * S, bool use_exceptions, unsigned num_threads);
 bool parse_commands(environment & env, io_state & ios, char const * fname, script_state * S,
                     bool use_exceptions, unsigned num_threads);
+void open_parser(lua_State * L);
 }
diff --git a/src/frontends/lean/parser_config.cpp b/src/frontends/lean/parser_config.cpp
index 4ba12986d5..bc18aa0310 100644
--- a/src/frontends/lean/parser_config.cpp
+++ b/src/frontends/lean/parser_config.cpp
@@ -75,6 +75,9 @@ serializer & operator<<(serializer & s, action const & a) {
     case action_kind::ScopedExpr:
         s << a.get_rec() << a.rbp() << a.use_lambda_abstraction();
         break;
+    case action_kind::LuaExt:
+        s << a.get_lua_fn();
+        break;
     case action_kind::Ext:
         lean_unreachable();
     }
@@ -104,6 +107,8 @@ action read_action(deserializer & d) {
         d >> rec >> rbp >> use_lambda_abstraction;
         return notation::mk_scoped_expr_action(rec, rbp, use_lambda_abstraction);
     }
+    case action_kind::LuaExt:
+        return notation::mk_ext_lua_action(d.read_string().c_str());
     case action_kind::Ext:
         break;
     }
diff --git a/src/frontends/lean/register_module.cpp b/src/frontends/lean/register_module.cpp
index 4a94ca7b75..e425f2d29b 100644
--- a/src/frontends/lean/register_module.cpp
+++ b/src/frontends/lean/register_module.cpp
@@ -9,11 +9,14 @@ Author: Leonardo de Moura
 #include "util/script_state.h"
 #include "frontends/lean/token_table.h"
 #include "frontends/lean/parse_table.h"
+#include "frontends/lean/parser.h"
 
 namespace lean {
+
 void open_frontend_lean(lua_State * L) {
     open_token_table(L);
     open_parse_table(L);
+    open_parser(L);
 }
 void register_frontend_lean_module() {
     script_state::register_module(open_frontend_lean);