Every function from a finite field to itself can be represented by a polynomial. The functions which are also permutations give rise to ``permutation polynomials," which have potential applications in cryptology. We will introduce a generalization of permutation polynomials called ``degree-preserving polynomials" and show a classification scheme of the latter. The criteria for a polynomial to qualify as degree preserving are certainly less stringent than those for the permuting qualification. Thus the idea to study degree-preserving polynomials allows more opportunity to maneuver and gain intuition about the occurrence of such polynomials.