  Star configurations are certain unions of linear subspaces of projective
space that have been studied extensively. We develop a framework for studying a
substantial generalization, which we call matroid configurations, whose ideals
generalize Stanley-Reisner ideals of matroids. Such a matroid configuration is
a union of complete intersections of a fixed codimension. Relating these to the
Stanley-Reisner ideals of matroids and using methods of Liaison Theory allows
us, in particular, to describe the Hilbert function and minimal generators of
the ideal of, what we call, a hypersurface configuration. We also establish
that the symbolic powers of the ideal of any matroid configuration are
Cohen-Macaulay. As applications, we study ideals coming from certain complete
hypergraphs and ideals derived from tetrahedral curves. We also consider
Waldschmidt constants and resurgences. In particular, we determine the
resurgence of any star configuration and many hypersurface configurations.
Previously, the only non-trivial cases for which the resurgence was known were
certain monomial ideals and ideals of finite sets of points. Finally, we point
out a connection to secant varieties of varieties of reducible forms.
