Abstract
The Tutte polynomial is a classical invariant of both graphs and matroids. In this paper, we give a survey of some motivations for the Tutte polynomial, its properties, and possible applications. We cover basic graph theory and graphical invariants and how they relate to the Tutte polynomial. We define matroid theory and extend the Tutte polynomial to matroids. Then we describe a generalization of the Tutte polynomial to polymatroids.