  Intrinsic volumes, which generalize both Euler characteristic and Lebesgue
volume, are important properties of $d$-dimensional sets. A random cubical
complex is a union of unit cubes, each with vertices on a regular cubic
lattice, constructed according to some probability model. We analyze and give
exact polynomial formulae, dependent on a probability, for the expected value
and variance of the intrinsic volumes of several models of random cubical
complexes. We then prove a central limit theorem for these intrinsic volumes.
For our primary model, we also prove an interleaving theorem for the zeros of
the expected-value polynomials. The intrinsic volumes of cubical complexes are
useful for understanding the shape of random $d$-dimensional sets and for
characterizing noise in applications.
