How many ways are there of covering a chess board with dominoes (the dimer problem) or covering a fraction p of the squares (the monomer-dimer problem). Dominoes cover two neighboring squares, and must not overlap. This easily generalizes to the same questions on arbitrary lattices in any dimension. We discuss results (mostly very new) , proved, numerical, conjectured, particularly for d-dimensional rectangular lattices. There is a wonderful conjecture. Certainly no familiarity with dimer problems is assumed.