A polynomial over a finite ring R is called permutation polynomial if it induces a bijection from R to R. Permutation polynomials have been a subject of study for many years and have applications in many areas of science and engineering. This monograph contains some results related to permutation polynomials over finite rings and finite fields. A survey of known results and detailed bibliography is provided. A special feature is the applications of permutation polynomials in multivariate public key cryptography. Multivariate public key cryptography is a branch of public key cryptography in which the cryptosystems are based on the problem of solving nonlinear equations over finite fields. This monograph contains design and analysis of some efficient multivariate public key cryptosystems using permutation polynomials over finite fields.