Giving Odds

The art of giving odds in a game of chess has been a long neglected art. Popular in the 19th century, it began to die out as the 20th century approached, possibly as a consequence of the introduction of chess clocks. Instead of spotting your opponent material, just give them a headstart on the clock. But I'm not convinced that the extra time helps the weaker player as much as the extra material does. And anyone who watched Greek GM Ioannis Papaioannou play 1 minute v 5 minutes and still destroy all comers at 3am during the Majorca Olympiad would probably agree with me.
One area where giving odds is very useful is when coaching. Instead of handing out a pasting to a weaker player, or trying to avoid playing your best, "to make a game of it", spotting your opponent a rook or a queen allows both players to try their best, while still maintaining an even level of competition.
You can even use it is a useful measure of progress, reducing the odds every time you drop a game to a student. I will usually start by offering queen odds, and then going rook, knight and finally pawn.

Here is one of the classic odds games in chess history. Paul Morphy starts without a Queens Rook and finishes the game by giving checkmate by castling.

Morphy,P - NN [C57]
New Orleans sim New Orleans (5), 1858
(Remove Whites Queen Rook)
1.e4 e5 2.Nf3 Nc6 3.Bc4 Nf6 4.Ng5 d5 5.exd5 Nxd5 6.Nxf7 Kxf7 7.Qf3+ Ke6 8.Nc3 Nd4 9.Bxd5+ Kd6 10.Qf7 Be6 11.Bxe6 Nxe6 12.Ne4+ Kd5 13.c4+ Kxe4 14.Qxe6 Qd4 15.Qg4+ Kd3 16.Qe2+ Kc2 17.d3+ Kxc1 18.O-O# 1-0