A mathematical definition of the concept of elementary mode is given so as to apply to biochemical reaction systems subsisting at steady state. This definition relates to existing concepts of null-space vectors and includes a condition of simplicity. It is shown that for systems in which all fluxes have fixed signs, all elementary modes are given by the generating vectors of a convex cone and can, thus, be computed by an existing algorithm. The present analysis allows for the more general case that some reactions can proceed in either direction. Basic ideas how to compute the complete set of elementary modes in this situation are outlined and verified by way of several examples, with one of them representing glycolysis and gluconeogenesis. These examples show that the elementary modes can be interpreted in terms of the particular biochemical functions of the network. The relationships to (futile) substrate cycles are elucidated.