  In this paper we consider coherent systems $(E,V)$ on an elliptic curve which
are stable with respect to some value of a parameter $\alpha$. We show that the
corresponding moduli spaces, if non-empty, are smooth and irreducible of the
expected dimenson. Moreover we give precise conditions for non-emptiness of the
moduli spaces. Finally we study the variation of the moduli spaces with
$\alpha$.
