By 1939 or during the year 1939, Italy was in possession of Sardinia, Albania, Sicily, the Dodecanese Islands, Libya and what was termed "Italian East Africa", consisting of today's Ethiopia, Eritrea and Somalia. Germany had annexed Austria in 1938 and in '39 Czechoslovakia (including Bohemia and Moravia), and of course in September invaded Poland.
Following these successes there was a third party added to the Axis the following year, the Japanese Empire, which in 1939 had ruled Manchuria (as Manchukuo), Korea, the Ryukuan Islands and a fairly large number of islands spread across the Pacific. It was the conquest of France by Germany in 1940 which both gave Japan confidence in the Axis Powers and convinced them to annex the French colonies of Indochina. That action led the United States to freeze Japanese economic assets in the US, leading to America's entry into WWII.
By the end of 1939, Hitler had already painlessly taken over Austria through the Anschluss (1938), whereby the country virtually joined Hitler's Third Reich of its own free will. He then took the Sudetenland from Czechoslovakia in late 1938 through an appeasement deal struck in Munich with Neville Chamberlain. In March of 1939, his armies took the rest of Czechoslovakia.
In September of 1939, he began World War II by attacking and conquering Poland. Italy, meanwhile, had taken control of Ethiopia from Emperor Haile Selassie in 1936, which is about as far as Mussolini's "empire" went.
If you are using a textbook for this, I would strongly advise you to consider the information in this source before all else. With that in mind, I would suggest that much of the European continent remained in Axis control at the point of 1939. In looking at a map of Europe, the Allied powers of England and Russia bookended a great deal of Axis control. Moving from West to East, France, Belgium, the Netherlands, Norway, Denmark, Hungary, Czechoslovakia, Finland, the Baltic Nations, Poland, Romania, and Bulgaria were all under allied control. Spain and Portugal were neutral, although the former was under the control of the dictator Franco, and Sweden, Switzerland, and Ireland were neutral.