A polynomial f over a finite field F is called a permutation
polynomial if the mapping f from F to F permutes the
elements of F. Permutation polynomials were first investigated
by Hermite, and since then, many studies concerning them have been
devoted. In the last 20 years there has been a revival in the
interest for permutation polynomials, in part due to their
cryptographic applications. In this talk I will give a brief
introduction to permutation polynomials over finite fields and
then describe new classes of permutation polynomials. In
particular, connections between a generalized Lucas sequence and
permutation binomials will be discussed. This is a joint work with
Amir Akbary.