  Throughout this paper we study the existence of irreducible curves C on
smooth projective surfaces S with singular points of prescribed topological
types S_1,...,S_r. There are necessary conditions for the existence of the type
\sum_{i=1}^r \mu(S_i) < aC^2+bC.K+c+1 for some fixed divisor K on S and
suitable coefficients a, b and c, and the main sufficient condition that we
find is of the same type, saying it is asymptotically optimal. Even for the
case where S is the projective plane, ten years ago general results of this
quality have not been known.
  An important ingredient for the proof is a vanishing theorem for invertible
sheaves on the blown up S of the form O_{S'}(\pi^*D-\sum_{i=1}^r m_iE_i),
deduced from the Kawamata-Vieweg Vanishing Theorem. Its proof covers the first
part of the paper, while the middle part is devoted to the existence theorems.
In the last part we investigate our conditions on ruled surfaces, products of
elliptic curves, surfaces in projective 3-space, and K3-surfaces.
