For the equi-affine group epsilon(n) of transformations of R-n, definitions of an epsilon(n)-equivalence of curves and an equi-affine type of a curve are introduced. The epsilon(n)-equivalence of curves is reduced to the problem of the epsilon(n)-equivalence of paths. A generating system of the differential ring of epsilon(n)-invariant differential polynomial functions of curves is described. Global conditions of the epsilon(n)-equivalence of curves are given in terms of the equi-affine type of a curve and the generating differential invariants. An independence of the generating differential invariants is proved. (C) 2003 Elsevier B.V. All rights reserved.