The orientable genus of a graph is the minimum number of handles needed to embed that graph on a surface. Determining the genus of a graph is a fundamental yet very difficult problem. In this paper we show that the orientable genus of a randomly selected graph on n vertices has a strong tendency towards n²/24. More strongly, we give upper and lower bounds on the genus which hold for almost all graphs. We extend our results to the graph model in which the edge-probability varies with the order of the graph and to the model in which the number of edges is preordained. We also give the analogous bounds for the nonorientable genus. Loosely speaking, a random graph almost always has embeddings on both orientable and nonorientable surfaces which are nearly triangulations.