I have noticed there are folks here with different historical backgrounds. Perhaps we could benefit each other by suggesting the best possible textbooks in our respective areas of specialty or interest.
For modern China, my favorite textbook is called 'The Search for Modern China' by Jonathan Spence. 'A History of Chinese Civilization' by Jacques Gernet (tr. from French) gives excellent coverage of the pre-modern era. For Japan, I haven't seen a book that surpasses 'Japan: Tradition and Transformation' by Reischauer and Craig, although Peter Duus's book 'Modern Japan' does give interesting coverage and insight especially on the last 100 years. 'The Golden Peninsula' by Charles Keyes is not a textbook, per se, but gives very balanced coverage of the various Southeast Asian cultures from an anthropological standpoint. For a more comprehensive but sometimes vague read, try 'In Search of Southeast Asia' (Steinberg, Ed.). Donald Stone Macdonald created the prototype text on modern Korea, entitled, strangely enough 'The Koreans'.
I have always enjoyed dabbling in ancient history topics. My favorite is 'Rome' by Rostovtzeff. For excellent coverage of the ancient to the modern period of Western history, Greer's 'A Brief History of the Western World' might do the trick.
Please add your favorite choices here!