diff --git a/clang-tools-extra/clangd/CodeComplete.cpp b/clang-tools-extra/clangd/CodeComplete.cpp
--- a/clang-tools-extra/clangd/CodeComplete.cpp
+++ b/clang-tools-extra/clangd/CodeComplete.cpp
@@ -452,18 +452,52 @@
   std::string summarizeSnippet() const {
     if (IsUsingDeclaration)
       return "";
-    // Suppress function argument snippets if args are already present.
-    if ((Completion.Kind == CompletionItemKind::Function ||
-         Completion.Kind == CompletionItemKind::Method ||
-         Completion.Kind == CompletionItemKind::Constructor) &&
-        NextTokenKind == tok::l_paren)
-      return "";
     auto *Snippet = onlyValue<&BundledEntry::SnippetSuffix>();
     if (!Snippet)
       // All bundles are function calls.
       // FIXME(ibiryukov): sometimes add template arguments to a snippet, e.g.
       // we need to complete 'forward<$1>($0)'.
       return "($0)";
+    // Suppress function argument snippets cursor is followed by left
+    // parenthesis (and potentially arguments) or if there are potentially
+    // template arguments. There are cases where it would be wrong (e.g. next
+    // '<' token is a comparison rather than template argument list start) but
+    // it is less common and suppressing snippet provides better UX.
+    if (Completion.Kind == CompletionItemKind::Function ||
+        Completion.Kind == CompletionItemKind::Method ||
+        Completion.Kind == CompletionItemKind::Constructor) {
+      // If there is a potential template argument list, drop snippet and just
+      // complete symbol name. Ideally, this could generate an edit that would
+      // paste function arguments after template argument list but it would be
+      // complicated. Example:
+      //
+      // fu^<int> -> function<int>
+      if (NextTokenKind == tok::less && Snippet->front() == '<')
+        return "";
+      // Potentially followed by argument list.
+      if (NextTokenKind == tok::l_paren) {
+        // If snippet contains template arguments we will emit them and drop
+        // function arguments. Example:
+        //
+        // fu^(42) -> function<int>(42);
+        if (Snippet->front() == '<') {
+          // Find matching '>'. Snippet->find('>') will not work in cases like
+          // template <typename T=std::vector<int>>. Hence, iterate through
+          // the snippet until the angle bracket balance reaches zero.
+          int Balance = 0;
+          size_t I = 0;
+          do {
+            if (Snippet->at(I) == '>')
+              --Balance;
+            else if (Snippet->at(I) == '<')
+              ++Balance;
+            ++I;
+          } while (Balance > 0);
+          return Snippet->substr(0, I);
+        }
+        return "";
+      }
+    }
     if (EnableFunctionArgSnippets)
       return *Snippet;
 
diff --git a/clang-tools-extra/clangd/unittests/CodeCompleteTests.cpp b/clang-tools-extra/clangd/unittests/CodeCompleteTests.cpp
--- a/clang-tools-extra/clangd/unittests/CodeCompleteTests.cpp
+++ b/clang-tools-extra/clangd/unittests/CodeCompleteTests.cpp
@@ -3114,10 +3114,13 @@
       Contains(AllOf(Labeled("Container<typename T>(int Size)"),
                      SnippetSuffix("<${1:typename T}>(${2:int Size})"),
                      Kind(CompletionItemKind::Constructor))));
-  // FIXME(kirillbobyrev): It would be nice to still produce the template
-  // snippet part: in this case it should be "<${1:typename T}>".
   EXPECT_THAT(
       completions(Context + "Container c = Cont^()", {}, Opts).Completions,
+      Contains(AllOf(Labeled("Container<typename T>(int Size)"),
+                     SnippetSuffix("<${1:typename T}>"),
+                     Kind(CompletionItemKind::Constructor))));
+  EXPECT_THAT(
+      completions(Context + "Container c = Cont^<int>()", {}, Opts).Completions,
       Contains(AllOf(Labeled("Container<typename T>(int Size)"),
                      SnippetSuffix(""),
                      Kind(CompletionItemKind::Constructor))));