We study minimum degree conditions for which a graph with given odd girth has a simple structure. For example, the classical work of Andrasfai, Erdos, and Sos implies that every n- vertex graph with odd girth 2k + 1 and minimum degree bigger than 2/2k+1 n must be bipartite. We consider graphs with a weaker condition on the minimum degree. Generalizing results of Haggkvist and of Haggkvist and Jin for the cases k = 2 and 3, we show that every n- vertex graph with odd girth 2k + 1 and minimum degree bigger than 3/4k n is homomorphic to the cycle of length 2k + 1. This is best possible in the sense that there are graphs with minimum degree 3/4k n and odd girth 2k + 1 that are not homomorphic to the cycle of length 2k + 1. Similar results were obtained by Brandt and Ribe- Baumann. (C) 2014 Wiley Periodicals, Inc.