We study one-parameter families of S-unit equations of the form f (t)u + g(t)v = h(t), where f , g, and h are univariate polynomials over a number field, t is an S-integer, and u and v are S-units. For many possible choices of f , g, and h, we are able to determine all but finitely many solutions to the corresponding one-parameter family of S-unit equations. The results are obtained as consequences of some recent results on integral points on surfaces.