Index: flang/lib/Semantics/check-declarations.cpp
===================================================================
--- flang/lib/Semantics/check-declarations.cpp
+++ flang/lib/Semantics/check-declarations.cpp
@@ -112,6 +112,7 @@
   }
   bool IsResultOkToDiffer(const FunctionResult &);
   void CheckBindC(const Symbol &);
+  void CheckBindCFunctionResult(const Symbol &);
   // Check functions for defined I/O procedures
   void CheckDefinedIoProc(
       const Symbol &, const GenericDetails &, GenericKind::DefinedIo);
@@ -360,7 +361,9 @@
       symbol.Rank() == 0) { // C830
     messages_.Say("CONTIGUOUS POINTER must be an array"_err_en_US);
   }
-  if (IsDummy(symbol)) {
+  bool dummy{IsDummy(symbol)};
+  bool funcRes{IsFunctionResult(symbol)};
+  if (dummy) {
     if (IsNamedConstant(symbol)) {
       messages_.Say(
           "A dummy argument may not also be a named constant"_err_en_US);
@@ -370,7 +373,7 @@
       messages_.Say(
           "A dummy argument may not have the SAVE attribute"_err_en_US);
     }
-  } else if (IsFunctionResult(symbol)) {
+  } else if (funcRes) {
     if (IsNamedConstant(symbol)) {
       messages_.Say(
           "A function result may not also be a named constant"_err_en_US);
@@ -380,6 +383,29 @@
       messages_.Say(
           "A function result may not have the SAVE attribute"_err_en_US);
     }
+    CheckBindCFunctionResult(symbol);
+  }
+  if (dummy || funcRes) {
+    if (innermostSymbol_ && IsBindCProcedure(*innermostSymbol_)) {
+      if (const DeclTypeSpec * type{symbol.GetType()};
+          type && type->category() == DeclTypeSpec::Character) {
+        bool isConstOne{false}; // 18.3.1(1)
+        if (const auto &len{type->characterTypeSpec().length().GetExplicit()}) {
+          if (auto constLen{evaluate::ToInt64(*len)}) {
+            isConstOne = constLen == 1;
+          }
+        }
+        if (!isConstOne) {
+          if (funcRes) {
+            messages_.Say(
+                "BIND(C) function result must have the length type parameter of value 1 if it is character type"_err_en_US);
+          } else {
+            messages_.Say(
+                "BIND(C) procedure argument must have the length type parameter of value 1 if it is character type"_err_en_US);
+          }
+        }
+      }
+    }
   }
   if (symbol.owner().IsDerivedType() &&
       (symbol.attrs().test(Attr::CONTIGUOUS) &&
@@ -397,6 +423,22 @@
 
 void CheckHelper::CheckCommonBlock(const Symbol &symbol) { CheckBindC(symbol); }
 
+void CheckHelper::CheckBindCFunctionResult(const Symbol &symbol) { // C1553
+  if (!innermostSymbol_ || !IsBindCProcedure(*innermostSymbol_)) {
+    return;
+  }
+  if (IsPointer(symbol) || IsAllocatable(symbol)) {
+    messages_.Say(
+        "BIND(C) function result cannot have ALLOCATABLE or POINTER attribute"_err_en_US);
+  }
+  if (symbol.Rank() > 0) {
+    messages_.Say("BIND(C) function result must be scalar"_err_en_US);
+  }
+  if (symbol.Corank()) {
+    messages_.Say("BIND(C) function result cannot be a coarray"_err_en_US);
+  }
+}
+
 void CheckHelper::CheckValue(
     const Symbol &symbol, const DerivedTypeSpec *derived) { // C863 - C865
   if (!IsDummy(symbol)) {
Index: flang/test/Semantics/bind-c09.f90
===================================================================
--- /dev/null
+++ flang/test/Semantics/bind-c09.f90
@@ -0,0 +1,57 @@
+! RUN: %python %S/test_errors.py %s %flang_fc1
+! Check for C1553 and 18.3.4(1)
+
+function func1() result(res) bind(c)
+  ! ERROR: BIND(C) function result cannot have ALLOCATABLE or POINTER attribute
+  integer, pointer :: res
+end
+
+function func2() result(res) bind(c)
+  ! ERROR: BIND(C) function result cannot have ALLOCATABLE or POINTER attribute
+  integer, allocatable :: res
+end
+
+function func3() result(res) bind(c)
+  ! ERROR: BIND(C) function result must be scalar
+  integer :: res(2)
+end
+
+function func4(arg) result(res) bind(c)
+  ! ERROR: BIND(C) function result must have the length type parameter of value 1 if it is character type
+  character(*) :: res
+  ! ERROR: BIND(C) procedure argument must have the length type parameter of value 1 if it is character type
+  character(*) :: arg
+end
+
+function func5(n, arg) result(res) bind(c)
+  integer :: n
+  ! ERROR: BIND(C) function result must have the length type parameter of value 1 if it is character type
+  character(n) :: res
+  ! ERROR: BIND(C) procedure argument must have the length type parameter of value 1 if it is character type
+  character(n) :: arg
+end
+
+function func6(arg) result(res) bind(c)
+  ! ERROR: BIND(C) function result must have the length type parameter of value 1 if it is character type
+  character(2) :: res
+  ! ERROR: BIND(C) procedure argument must have the length type parameter of value 1 if it is character type
+  character(2) :: arg
+end
+
+function func7(arg) result(res) bind(c)
+  integer, parameter :: n = 1
+  character(n) :: res, arg! OK
+end
+
+function func8(arg) result(res) bind(c)
+  ! ERROR: BIND(C) function result cannot have ALLOCATABLE or POINTER attribute
+  ! ERROR: BIND(C) function result must have the length type parameter of value 1 if it is character type
+  character(:), pointer :: res
+  ! ERROR: BIND(C) procedure argument must have the length type parameter of value 1 if it is character type
+  character(:), pointer :: arg
+end
+
+function func9() result(res) bind(c)
+  ! ERROR: BIND(C) function result cannot be a coarray
+  integer :: res[10, *]
+end