Follow-up question: What about the rest of the Allies and Axis countries?
Another follow-up question: Any good books on the U.S. Army in the WW2 period?
In regards to your second follow-up question in regards to good books on the United States Army in WWII, Rick Atkinson's Liberation Trilogy tells the story of the Army, along with small looks at other Allied forces in North Africa during 1942 and 1943 (An Army at Dawn), Sicily and the Italian Campaign up to the liberation of Rome (The Day of Battle), and Western Europe (The Guns at Last Light). Stephen E. Ambrose's book "Citizen Soldiers" is a more general tale of the entire Army's experience in Western Europe.
If you're looking for books about specific branches of the Army, Harry Yeide's book "Steel Victory" (of which there are two editions) tells the story of the separate tank battalions. Another book by Yeide, "The Tank Killers", is about the Tank Destroyer force. Yeide also has another excellent book, "Steeds of Steel", about the Army's mechanized cavalry troops.
The United States Army's official history of itself during the WWII period can be found here, along with other period documents;