  Permutation polynomials and their compositional inverses have wide
applications in cryptography, coding theory, and combinatorial designs.
Motivated by several previous results on finding compositional inverses of
permutation polynomials of different forms, we propose a general method for
finding these inverses of permutation polynomials constructed by the AGW
criterion. As a result, we have reduced the problem of finding the
compositional inverse of such a permutation polynomial over a finite field to
that of finding the inverse of a bijection over a smaller set. We demonstrate
our method by interpreting several recent known results, as well as by
providing new explicit results on more classes of permutation polynomials in
different types. In addition, we give new criteria for these permutation
polynomials being involutions. Explicit constructions are also provided for all
involutory criteria.
