This book addresses certain ways of representing sporadic simple groups and the graphs associated with these representations. It provides descriptions of all permutation representations of rank up to five of the sporadic groups and gives an introduction to practical computational techniques required for studying finite vertex-transitive graphs. The authors offer sufficient information about most of the groups to enable readers to reconstruct and study these representations themselves. Elementary techniques for analysis of finite vertex-transitive graphs are carefully explained, making this book not only an essential reference for experts, but also an excellent starting point for graduate students and researchers in this area.