In chess, the bishop and knight checkmate is the checkmate of a lone king by an opposing king, knight, and bishop. With the stronger side to move, checkmate can be forced in at most thirty-three moves from almost any starting position. Although it is classified as one of the four basic checkmates, the bishop and knight checkmate occurs in practice only approximately once in every 6,000 games.

A method for checkmate applicable when the lone king is in the corner of the opposite color from the bishop (the "wrong" corner, where checkmate cannot be forced), was given by François-André Danican Philidor in the 1777 update to his famous 1749 treatise, L'Analyse des Échecs. He called attention to the route of the knight now known as the "W manoeuvre".

Another method, known as "Delétang's Method" or "Delétang's Triangles", applicable when the lone king is unable to reach the longest diagonal of the color opposite to that of the bishop, involves confining the lone king in a series of three increasingly smaller triangles, ultimately forcing it into a corner of the same color as the bishop (the "right" corner). Some of the ideas of this method date back to 1780, but the complete system was first published in 1923 by Daniel Delétang. The method as propounded is not optimal, but it is relatively simple; so long as White has trapped the king behind the diagonal in a reasonable number of moves, it will lead to mate before the fifty-move rule takes effect.

In 1983, Julius Telesin showed that a king, bishop, and knight can force checkmate on the lone enemy king on an arbitrarily large board, as long as it contains a corner of the color that the bishop travels on. While the other basic checkmates can be forced in O(n) moves on an n × n board, Telesin's method gives an O(n²) bound for a bishop and knight checkmate. It has not been proven whether this is optimal.

Opinions differ among chess authors as to whether or not a player should learn this checkmate procedure.

Jeremy Silman omitted the bishop-and-knight checkmate from his Complete Endgame Course, claiming he had encountered it only once, and that his friend John Watson had never encountered it. Silman said, "Mastering it would take a significant chunk of time. Should the chess hopeful really spend many of his precious hours he's put aside for chess study learning an endgame he will achieve (at most) only once or twice in his lifetime?" Similarly, Grandmaster Jonathan Hawkins reported only ever encountering the position in a game once.

On the other hand, while grandmaster Andy Soltis concedes that he has never played this endgame and most players will never have it in their career, he argues that learning the checkmate teaches techniques that can be applied elsewhere. James Howell includes the bishop-and-knight checkmate in his book, saying that he has defended against it three times and that it occurs more often than the checkmate with two bishops; he omits the latter from his book. Finally, the checkmate occurred in at least one very notable case: Tal Shaked's victory over Alexander Morozevich in the penultimate round of the 1997 World Junior Chess Championship. Shaked knew the correct mating pattern, and his victory catapulted him to becoming World Junior Champion, whereas a draw would have prevented him from winning the title.

This section is adapted from Yasser Seirawan's Winning Chess Endings. It is assumed that White has the bishop and knight.