Question marks end all direct questions. This includes incomplete questions and statements intended as questions.
Direct Question: What is your name?
Incomplete Question: Really? When? No kidding?
Statement Intended as Question: Your name is Fred?
Sentences which describe a question but do not directly ask a question are called indirect questions. They do not take a question mark.
Incorrect: He asked if he could leave early?
(Describes but does not ask a question)
Correct: He asked if he could leave early.
Correct: He asked, "May I leave early?"
(In the last one, the question is directly quoted.)
Use a question mark in parentheses after a point of fact to show uncertainty about it. Use sparingly and only for items impossible to verify.
Example: His great-great-grandfather (Nelson Bridger?) supposedly fought in the Black Hawk War.
Example: Chaucer was born in 1343 (?).
(Note that a question mark used this way is not an end mark. A period is still needed.)
See also Question Marks in Quotations