We present a general technique for obtaining permutation polynomials over a finite field from permutations of a subfield. By applying this technique to the simplest classes of permutation polynomials on the subfield, we obtain several new families of permutation polynomials. Some of these have the additional property that both f(x) and f(x)+x induce permutations of the field, which has combinatorial consequences. We use some of our permutation polynomials to exhibit complete sets of mutually orthogonal latin squares. In addition, we solve the open problem from a recent paper by Wu and Lin, and we give simpler proofs of much more general versions of the results in two other recent papers.