This paper is a survey of distance-regular Cayley graphs of diameter two and three, primarily focusing on diameter three (due to the extensive amount of research already done in diameter two). The graphs were classified by their properties and related eigenvalues and multiplicities. Each classification was studied in depth to determine whether certain patterns held. Though many of the graphs could be formed in abelian groups, some were found to be genuinely non-abelian. Of special interest were the graphs formed by the dihedral and cyclic groups. In addition the existence of a (400,21,2,1
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