  We consider some variations on the classical method of Runge for effectively
determining integral points on certain curves. We first prove a version of
Runge's theorem valid for higher-dimensional varieties, generalizing a uniform
version of Runge's theorem due to Bombieri. We then take up the study of how
Runge's method may be expanded by taking advantage of certain coverings. We
prove both a result for arbitrary curves and a more explicit result for
superelliptic curves. As an application of our method, we solve certain
equations involving squares in products of terms in an arithmetic progression.
