Vs, teR § ~sO~t=~s+t . This notation is a continuous analogue of the concept of sequence of iterates of a map fe Hol(X, X); indeed, a sequence of iterates can be characterized as a map (P: N-~ --~ Hol (X, X) such that ~0 = idx and satisfying (1) for every s, t ~ N. The first paper concerning one-parameter semigroups of holomorphic maps seems to be [T], where problems somehow regarding the asymptotic behavior of one-parameter semigroups on 4, the unit disk in C, are studied. Later on, the typical approach used to be via the idea of fractional iteration; loosely stated, one wants to find a sensible way of defining, at least locally, the r-th iterate of a holomorphic function for any positive real number r. For a recent work on this subject, see [C]. The real break-through in the study of one-parameter semigroups in one complex variable is due to BERKSON and PORTA [BP] and HEINS [H]. Following [W2], they related semigroups and the theory of ordinary differential equations, being able to classify all one-parameter semigroups on Riemann surfaces (for a unified account of their results see [A3]). Strangely, there seems to be almost no papers on semigroups in several complex variables; as far as we know, they have been studied only in [A1, 2] and [V]. In this paper we want to generalize to arbitrary complex manifolds some of the results of [BP]; in particular, we want to describe at some extent the relationships between semigroups and ODE in several complex variables. First of all, we fix some notations. Let X and Y be two complex manifolds. A sequence {f, } c Hol (X, Y) is said compactly divergent if for every pair of compact sets