diff --git a/lib/templatesimplifier.cpp b/lib/templatesimplifier.cpp
index 021152220..22d66674f 100644
--- a/lib/templatesimplifier.cpp
+++ b/lib/templatesimplifier.cpp
@@ -240,11 +240,20 @@ unsigned int TemplateSimplifier::templateParameters(const Token *tok)
         if (Token::Match(tok, "& ::| %var%"))
             tok = tok->next();
 
+        // Skip '='
+        if (Token::Match(tok, "="))
+            tok = tok->next();
+        if (!tok)
+            return 0;
+
         // skip std::
         if (tok && tok->str() == "::")
             tok = tok->next();
-        while (Token::Match(tok, "%var% ::"))
+        while (Token::Match(tok, "%var% ::")) {
             tok = tok->tokAt(2);
+            if (tok->str() == "*") // Ticket #5759: Class member pointer as a template argument; skip '*'
+                tok = tok->next();
+        }
         if (!tok)
             return 0;
 
@@ -508,29 +517,51 @@ void TemplateSimplifier::useDefaultArgumentValues(const std::list<Token *> &temp
         //     x = y
         // this list will contain all the '=' tokens for such arguments
         std::list<Token *> eq;
+        // and this set the position of parameters with a default value
+        std::set<std::size_t> defaultedArgPos;
 
         // parameter number. 1,2,3,..
         std::size_t templatepar = 1;
 
+        // parameter depth
+        std::size_t templateParmDepth = 0;
+
         // the template classname. This will be empty for template functions
         std::string classname;
 
         // Scan template declaration..
         for (Token *tok = *iter1; tok; tok = tok->next()) {
+            if (Token::simpleMatch(tok, "template < >")) { // Ticket #5762: Skip specialization tokens
+                tok = tok->tokAt(2);
+                continue;
+            }
+
+            if (tok->str() == "<" && templateParameters(tok))
+                ++templateParmDepth;
+
             // end of template parameters?
-            if (tok->str() == ">") {
-                if (Token::Match(tok, "> class|struct %var%"))
+            if (tok->str() == ">" || tok->str() == ">>") {
+                if (Token::Match(tok, ">|>> class|struct %var%"))
                     classname = tok->strAt(2);
-                break;
+                templateParmDepth -= (1 + (tok->str() == ">>"));
+                if (0 == templateParmDepth)
+                    break;
             }
 
             // next template parameter
             if (tok->str() == ",")
                 ++templatepar;
 
-            // default parameter value
-            else if (tok->str() == "=")
-                eq.push_back(tok);
+            // default parameter value?
+            else if (Token::Match(tok, "= !!>")) {
+                if (defaultedArgPos.insert(templatepar).second) {
+                    eq.push_back(tok);
+                } else {
+                    // Ticket #5605: Syntax error (two equal signs for the same parameter), bail out
+                    eq.clear();
+                    break;
+                }
+            }
         }
         if (eq.empty() || classname.empty())
             continue;
diff --git a/test/testsimplifytokens.cpp b/test/testsimplifytokens.cpp
index 46cb6f4c3..d24e7d3a5 100644
--- a/test/testsimplifytokens.cpp
+++ b/test/testsimplifytokens.cpp
@@ -2461,10 +2461,10 @@ private:
         }
         {
             const char code[] = "template<class T, class T2 = A<T>> class B {};\n"
-                                "template<class B = A, typename C = C<B>> class C;\n"
+                                "template<class B = A, typename C = C<B>> class C {};\n"
                                 "template<class B, typename C> class D { };\n";
             ASSERT_EQUALS("template < class T , class T2 > class B { } ; "
-                          "template < class B , typename C > class C ; "
+                          "template < class B , typename C > class C { } ; "
                           "template < class B , typename C > class D { } ;", tok(code));
         }
     }
diff --git a/test/testtokenize.cpp b/test/testtokenize.cpp
index 064f70638..5676a13a6 100644
--- a/test/testtokenize.cpp
+++ b/test/testtokenize.cpp
@@ -417,6 +417,7 @@ private:
         TEST_CASE(syntax_error);
         TEST_CASE(syntax_error_templates_1);
         TEST_CASE(syntax_error_templates_2);
+        TEST_CASE(syntax_error_templates_3); // Ticket #5605 - invalid template declaration
 
         TEST_CASE(removeKeywords);
 
@@ -6503,6 +6504,20 @@ private:
         tokenizer.tokenize(istr, "test.cpp");   // shouldn't segfault
     }
 
+    void syntax_error_templates_3() { // Ticket #5605, #5759, #5762
+        tokenizeAndStringify("foo() template<typename T1 = T2 = typename = unused, T5 = = unused> struct tuple Args> tuple<Args...> { } main() { foo<int,int,int,int,int,int>(); }");
+        tokenizeAndStringify("( ) template < T1 = typename = unused> struct Args { } main ( ) { foo < int > ( ) ; }");
+        tokenizeAndStringify("() template < T = typename = x > struct a {} { f <int> () }");
+        tokenizeAndStringify("template < T = typename = > struct a { f <int> }");
+        tokenizeAndStringify("struct S { int i, j; }; "
+                             "template<int S::*p, typename U> struct X {}; "
+                             "X<&S::i, int> x = X<&S::i, int>(); "
+                             "X<&S::j, int> y = X<&S::j, int>(); ");
+        tokenizeAndStringify("template <typename T> struct A {}; "
+                             "template <> struct A<void> {}; "
+                             "void foo(const void* f = 0) {}");
+    }
+
     void removeKeywords() {
         const char code[] = "if (__builtin_expect(!!(x), 1));";