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With the exception of the pawns, which only move forward one square at a time (or two on the first move), chess pieces can move in any direction providing it is a legal move.
The king can move 1 square in any direction, but may not move into a square attacked by an enemy piece, nor capture a piece such that he is then attacked. When castling, the king moves two squares toward the rook, but may not land on or cross a square that is attacked by an enemy piece.
The queen moves any number of unblocked squares in any single direction : forward, backward, or diagonally. But not if her movement would expose the king to a direct attack. (A queen trapped between a queen, rook or bishop and her king is said to be "pinned".)
Until a pawn is promoted to a queen, rook, bishop or knight, it cannot move backwards.
Knights can move in any direction as long as there is an appropriate space available.