  For any given finitely presented group G there exists a closed oriented
4-manifold with fundamental group G. What pairs of integers can occur as the
signature and Euler characteristic of such a 4-manifold? The first part of this
paper develops general theory related to this question and applies this theory
to resolve the question for a large collection of groups, including surface
groups and 3-manifold groups. The second part of the paper offers an extended
analysis in the case of free abelian groups, G = Z^n. The paper concludes with
a list of open problems related to the general question.
