In 1930 Keller conjectured that every tiling of R^n by cubes had to have a certain amount of nice overlap between adjacent cubes. Since about 1940 the conjecture was known to be true in dimensions <7. Nevertheless, in 1992 the first counterexample was discovered in dimension 12 (and by an easy corollary in all higher dimensions) by a slight modification of a near-counterexample in dimension 3. In this talk I will discuss the problem and its connection to graph theory which has subsequently shown that it is false in dimension 8 and higher, leaving dimension 7 as the only unresolved case.