  We apply the idea of a topological quantum field theory (TQFT) to maps from
manifolds into topological spaces. This leads to a notion of a
(d+1)-dimensional homotopy quantum field theory (HQFT) which may be described
as a TQFT for closed d-dimensional manifolds and (d+1)-dimensional cobordisms
endowed with homotopy classes of maps into a given space. For a group $\pi$, we
introduce cohomological HQFT's with target $K(\pi,1)$ derived from cohomology
classes of $\pi$ and its subgroups of finite index. The main body of the paper
is concerned with (1+1)-dimensional HQFT's. We classify them in terms of so
called crossed group-algebras. In particular, the cohomological
(1+1)-dimensional HQFT's over a field of characteristic 0 are classified by
simple crossed group-algebras. We introduce two state sum models for
(1+1)-dimensional HQFT's and prove that the resulting HQFT's are direct sums of
rescaled cohomological HQFT's. We also discuss a version of the Verlinde
formula in this setting.
