  We classify four-dimensional compact solvmanifolds up to diffeomorphism,
while determining which of them have complex analytic structures. In
particular, we shall see that a four-dimensional compact solvmanifold S can be
written, up to double covering, as G/L where G is a simply connected solvable
Lie group and L is a lattice of G, and every complex structure J on S is the
canonical complex structure induced from a left-invariant complex structure on
G. We also give a complete list of all the complex structures on
four-dimensional compact homogeneous spaces, referring to their corresponding
complex surfaces.
