This paper examines the spread of diseases within populations in the context of networks of potentially disease-causing contacts. We examine the assumptions underlying classical mathematical models of epidemics and how more realistic assumptions can be made using contact networks. Several well-known kinds of contact networks are examined and simulated by evaluating their structural properties relevant to disease propagation. Algorithms used in the study of these networks are explained and numerical simulations of percolation and the epidemic process carried out to explore the effects that the network structure has on disease progression.