Checkmate is the goal of every chess game. The king is under attack (check) and there is no legal way to save it: the king cannot move to a safe square, the attack cannot be blocked, and the attacker cannot be captured. The game ends immediately, and the checkmated player loses.
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In notation, checkmate is written with a #, as in Ra8#.
The king is never actually captured. The game stops the moment checkmate happens.
The fastest possible checkmate is Fool’s Mate, which takes two moves: 1. f3 e5 2. g4 Qh4#.