  Star configurations are certain unions of linear subspaces of projective
space. They have appeared in several different contexts: the study of extremal
Hilbert functions for fat point schemes in the plane; the study of secant
varieties of some classical algebraic varieties; the study of the resurgence of
projective schemes. In this paper we study some algebraic properties of the
ideals defining star configurations, including getting partial results about
Hilbert functions, generators and minimal free resolutions of the ideals and
their symbolic powers. We also show that their symbolic powers define
arithmetically Cohen-Macaulay subschemes and we obtain results about the
primary decompositions of the powers of the ideals. As an application, we
compute the resurgence for the ideal of the codimension $n-1$ star
configuration in $\pr{n}$ in the monomial case (i.e., when the number of
hyperplanes is $n+1$).
