Abstract : The framework of this thesis is the equivariant theory of curves, i.e. the study of curves provided with an action of a group G, which is always supposed to be finite. The main result is an equivariant Riemann-Roch theorem with values in the character ring of the fixed group, and which lifts the usual theorem. It is obtained for G-sheaves of any rank thanks to the introduction of a group of divisors with equivariant coefficients which makes it possible in particular to define the determinant and the degree of such a sheaf. One applies this theorem to the computation of geometric Galois structures.