The main purpose of this thesis is an investigation of a class of graphs defined by polynomials over finite fields. There is a close relationship between these graphs and finite incidence structures such as projective planes and generalized quadrangles. Polynomial graphs also play an important role in extremal graph theory, and often questions on polynomial graphs lead to problems on permutation polynomials. Polynomial graphs of dimension 2 are related to projective planes. For a subclass of these graphs, an isomorphism criterion is given in Chapter 4. Polynomial graphs of dimension 3 are closely connected to generalized quadrangles. They are considered in Chapter 5; the main results characterize the existence of short cycles for several subclasses of these graphs. Open problems of extremal graph theory motivated studies of polynomial graphs of dimension 4; they are considered in Chapter 6. Often problems considered in Chapters 4–5 are related to the theory of permutation polynomials; they were studied in Chapter 3. The methods of investigations use different techniques from graph theory, geometry and the theory of finite fields as well as computational software.