In a previous article the authors showed that almost all labelled cubic graphs are hamiltonian. In the present article, this result is used to show that almost all r‐regular graphs are hamiltonian for any fixed r ⩾ 3, by an analysis of the distribution of 1‐factors in random regular graphs. Moreover, almost all such graphs are r‐edge‐colorable if they have an even number of vertices. Similarly, almost all r‐regular bipartite graphs are hamiltonian and r‐edge‐colorable for fixed r ⩾ 3. © 1994 John Wiley & Sons, Inc.