A Latin square is an nxn matrix with the numbers 1 through n appearing exactly once in every row and column. Latin squares have been the subject of many conjectures for over 200 years, the most famous of which is by Euler. Euler's conjecture turned out to be entirely wrong, and its disproof was reported on the front page of the New York Times in 1959. I will discuss several results and conjectures about Latin squares, as well as their relationship with finite projective planes.