We will focus on Beck’s conjecture that the chromatic number of a zero-divisor graph of a ring R is equal to the clique number of the ring R. We begin by calculating the chromatic number of the zero-divisor graphs for some finite rings and characterizing rings whose zero-divisor graphs have finite chromatic number, known as colorings. We will discuss some properties of colorings and elements called separating elements, which will allow us to determine that Beck’s conjecture holds for rings that are principal ideal rings and rings that are reduced. Then we will characterize the finite rings whose zero-divisor graphs have chromatic number less than or equal to four. In the general case, we will discuss a local ring that serves as a counterexample to Beck’s conjecture.