  In this paper we describe the Seiberg-Witten invariants, which have been
introduced by Witten, for manifolds with $b_+=1$. In this case the invariants
depend on a chamber structure, and there exists a universal wall crossing
formula. We take into account the contribution of the 1-homology of the
base-manifold. For every K\"ahler surface with $p_g=0$ and $q$=0, these
invariants are non-trivial for all $Spin^c(4)$-structures of non-negative
index.
