An Allumwandlung (German for "complete promotion", sometimes shortened to AUW) is a chess problem where, at some stage in the solution, a pawn (or sometimes pawns) is promoted variously to a knight, bishop, rook and queen.
Here is an example composed by Niels Hoeg and first published in 1905. White to move and mate in three:
The key move (white's first move) is f7 (see algebraic notation), and depending on how black defends, white promotes to either a queen, a rook, a bishop or a knight on move two. The lines are:
The importance of white's underpromotions can be understood by considering what happens if he promotes to a queen no matter what black plays: after 1... exf4 or 1... exd4 2. f8Q is stalemate, while following 1... Kf6 2. f8Q Kxg6 there is no mate.