  We define and study homotopy groups of cubical sets. To this end, we give
four definitions of homotopy groups of a cubical set, prove that they are
equivalent, and further that they agree with their topological analogues via
the geometric realization functor. We also provide purely combinatorial proofs
of several classical theorems, including: product preservation, commutativity
of higher homotopy groups, the long exact sequence of a fibration, and
Whitehead's theorem. This is a companion paper to our "Cubical setting for
discrete homotopy theory, revisited" in which we apply these results to study
the homotopy theory of simple graphs.
