Batter hits fair fly ball. A fielder throws his glove up, knocking the ball out of the air, and catches the ball before it hits the ground. Is the batter out?

Answer: No, the batter is not out.

Under Official Baseball Rules Rule 7.05 Each runner including the batter-runner may, without liability to be put out, advance—
(a) To home base, scoring a run, if a fair ball goes out of the playing field in flight and he touched all bases legally; or if a fair ball which, in the umpire’s judgment, would have gone out of the playing field in flight, is deflected by the act of a fielder in throwing his glove, cap, or any article of his apparel;
(b) Three bases, if a fielder deliberately touches a fair ball with his cap, mask or any part of his uniform detached from its proper place on his person. The ball is in play and the batter may advance to home base at his peril;
(c) Three bases, if a fielder deliberately throws his glove at and touches a fair ball. The ball is in play and the batter may advance to home base at his peril.
(d) Two bases, if a fielder deliberately touches a thrown ball with his cap, mask or any part of his uniform detached from its proper place on his person. The ball is in play;
(e) Two bases, if a fielder deliberately throws his glove at and touches a thrown ball. The ball is in play;
Rule 7.05(b) through 7.05(e) Comment: In applying (b-c-d-e) the umpire must rule that the thrown glove or detached cap or mask has touched the ball. There is no penalty if the ball is not touched. Under (c-e) this penalty shall not be invoked against a fielder whose glove is carried off his hand by the force of a batted or thrown ball, or when his glove flies off his hand as he makes an obvious effort to make a legitimate catch.