Let $D$ be a fixed set of prime numbers. In this thesis we investigate the chromatic number of graphs with vertex set of the integers and edges between any pair of vertices whose distance is in $D$. Such a graph is called a prime distance graph, and the set $D$ is called the distance set. The chromatic number of prime distance graphs is known when the distance set $D$ has at most four primes. In this thesis we begin to classify prime distance graphs with a distance set of five primes. The number theoretic function $\kappa (D)$ is used as a tool, and some general lemmas about $\kappa (D)$ are established.