In this paper we shall derive a concise formula for the number of Euler graphs on n labelled nodes and k edges. An Euler graph is a connected graph in which every node has even valency, where by the valency of a node is meant the number of edges which are incident with that node. Throughout most of the paper we shall be dealing with graphs whose nodes have even valencies but which may or may not be connected. For convenience we shall refer to these graphs as Euler graphs, although the usage is not, strictly speaking, correct. We shall impose the condition of connectedness in § 4.