diff --git a/src/cython_cmake/cmake/UseCython.cmake b/src/cython_cmake/cmake/UseCython.cmake
index feac051..4daf8bf 100644
--- a/src/cython_cmake/cmake/UseCython.cmake
+++ b/src/cython_cmake/cmake/UseCython.cmake
@@ -8,13 +8,13 @@
 # using cython.
 #
 #   Cython_compile_pyx(<pyx_file1> [<pyx_file2> ...]
-#                     [TARGET_LANGUAGE C | CXX]
+#                     [LANGUAGE C | CXX]
 #                     [LANGUAGE_LEVEL 2 | 3 | 3str]
-#                     [OUTPUT_VAR <OutputVar>])
+#                     [OUTPUT_VARIABLE <OutputVariable>])
 #
 # Options:
 #
-# ``TARGET_LANGUAGE [C | CXX]``
+# ``LANGUAGE [C | CXX]``
 #   Force the generation of either a C or C++ file. By default, a C file is
 #   generated, unless the C language is not enabled for the project; in this
 #   case, a C++ file is generated by default.
@@ -28,14 +28,17 @@
 #   Using the Python-3str syntax and semantics means there cython does not assume
 #   unicode is used by by default for string literals under Python 2
 #
-# ``OUTPUT_VAR <OutputVar>``
-#   Set the variable ``<OutputVar>`` in the parent scope to the path to the
+# ``CYTHON_ARGS <args>``
+#   Specify additional arguments for the cythonization process.
+#
+# ``OUTPUT_VARIABLE <OutputVariable>``
+#   Set the variable ``<OutputVariable>`` in the parent scope to the path to the
 #   generated source file.  By default, ``<Name>`` is used as the output
 #   variable name.
 #
 # Defined variables:
 #
-# ``<OutputVar>``
+# ``<OutputVariable>``
 #   The path of the generated source file.
 #
 # Usage example:
@@ -45,7 +48,7 @@
 #   find_package(Cython)
 #
 #   Cython_compile_pyx(_hello.pyx
-#     OUTPUT_VAR _hello_source_files
+#     OUTPUT_VARIABLE _hello_source_files
 #   )
 #
 #   Python_add_library(_hello
@@ -162,10 +165,10 @@ function(add_cython_target _name)
   mark_as_advanced(CYTHON_ANNOTATE CYTHON_FLAGS)
 
   if(_args_C)
-    set(_target_language "C")
+    set(_language "C")
   endif()
   if(_args_CXX)
-    set(_target_language "CXX")
+    set(_language "CXX")
   endif()
 
   if(_args_PY2)
@@ -215,10 +218,10 @@ function(add_cython_target _name)
   endif()
 
   Cython_compile_pyx(
-    TARGET_LANGUAGE ${_target_language}
+    LANGUAGE ${_language}
     LANGUAGE_LEVEL ${_language_level}
     CYTHON_ARGS ${_cython_args}
-    OUTPUT_VAR ${_args_OUTPUT_VAR}
+    OUTPUT_VARIABLE ${_args_OUTPUT_VAR}
     ${_source_file}
     )
 
@@ -229,7 +232,7 @@ endfunction()
 
 function(Cython_compile_pyx)
   set(_options )
-  set(_one_value LANGUAGE_LEVEL TARGET_LANGUAGE OUTPUT_VAR)
+  set(_one_value LANGUAGE_LEVEL LANGUAGE OUTPUT_VARIABLE)
   set(_multi_value CYTHON_ARGS)
 
   cmake_parse_arguments(_args
@@ -244,26 +247,26 @@ function(Cython_compile_pyx)
 
   # Set target language
   get_property(_languages GLOBAL PROPERTY ENABLED_LANGUAGES)
-  set(_target_language ${_args_TARGET_LANGUAGE})
-  if(NOT _target_language)
+  set(_language ${_args_LANGUAGE})
+  if(NOT _language)
     if("C" IN_LIST _languages)
-      set(_target_language "C")
+      set(_language "C")
     elseif("CXX" IN_LIST _languages)
-      set(_target_language "CXX")
+      set(_language "CXX")
     endif()
   else()
-    if(NOT _target_language MATCHES "^(C|CXX)$")
-      message(FATAL_ERROR "TARGET_LANGUAGE must be one of C or CXX")
+    if(NOT _language MATCHES "^(C|CXX)$")
+      message(FATAL_ERROR "LANGUAGE must be one of C or CXX")
     endif()
   endif()
 
-  set(_target_language_C_arg "")
-  set(_target_language_C_extension "c")
-  set(_target_language_CXX_arg "--cplus")
-  set(_target_language_CXX_extension "cxx")
+  set(_language_C_arg "")
+  set(_language_C_extension "c")
+  set(_language_CXX_arg "--cplus")
+  set(_language_CXX_extension "cxx")
 
-  set(_target_language_arg ${_target_language_${_target_language}_arg})
-  set(_target_language_extension ${_target_language_${_target_language}_extension})
+  set(_language_arg ${_language_${_language}_arg})
+  set(_language_extension ${_language_${_language}_extension})
 
   # Set language level
   set(_language_level ${_args_LANGUAGE_LEVEL})
@@ -290,13 +293,13 @@ function(Cython_compile_pyx)
 
   foreach(_source_file IN LISTS _source_files)
     cmake_path(GET _source_file STEM _name)
-    set(generated_file "${CMAKE_CURRENT_BINARY_DIR}/${_name}.${_target_language_extension}")
+    set(generated_file "${CMAKE_CURRENT_BINARY_DIR}/${_name}.${_language_extension}")
     set_source_files_properties(${generated_file} PROPERTIES GENERATED TRUE)
 
     file(RELATIVE_PATH generated_file_relative
         ${CMAKE_BINARY_DIR} ${generated_file})
 
-    set(comment "Generating ${_target_language} source ${generated_file_relative}")
+    set(comment "Generating ${_language} source ${generated_file_relative}")
 
     get_source_file_property(pyx_location ${_source_file} LOCATION)
 
@@ -305,7 +308,7 @@ function(Cython_compile_pyx)
       OUTPUT ${generated_file}
       COMMAND ${CYTHON_EXECUTABLE}
       ARGS
-        ${_target_language_arg}
+        ${_language_arg}
         ${_language_level_arg}
         ${_args_CYTHON_ARGS}
         ${_depfile_arg}
@@ -321,9 +324,9 @@ function(Cython_compile_pyx)
     list(APPEND generated_files ${generated_file})
   endforeach()
 
-  if(_args_OUTPUT_VAR)
-    set(_output_var ${_args_OUTPUT_VAR})
-    set(${_output_var} ${generated_files} PARENT_SCOPE)
+  if(_args_OUTPUT_VARIABLE)
+    set(_output_variable ${_args_OUTPUT_VARIABLE})
+    set(${_output_variable} ${generated_files} PARENT_SCOPE)
   endif()
 
 endfunction()
diff --git a/tests/packages/simple/CMakeLists.txt b/tests/packages/simple/CMakeLists.txt
index af8aebd..4d7f134 100644
--- a/tests/packages/simple/CMakeLists.txt
+++ b/tests/packages/simple/CMakeLists.txt
@@ -7,7 +7,7 @@ find_package(
   REQUIRED)
 find_package(Cython MODULE REQUIRED VERSION 3.0)
 
-cython_compile_pyx(simple.pyx TARGET_LANGUAGE C OUTPUT_VAR simple_c)
+cython_compile_pyx(simple.pyx LANGUAGE C OUTPUT_VARIABLE simple_c)
 
 python_add_library(simple MODULE "${simple_c}" WITH_SOABI)