  We prove the existence of solutions for the Monge minimization problem,
addressed in a metric measure space $(X,d,m)$ enjoying the Riemannian
curvature-dimension condition $\RCD(K,N)$, with $N < \infty$. For the first
marginal measure, we assume that $\mu_{0} \ll m$. As a corollary, we obtain
that the Monge problem and its relaxed version, the Monge-Kantorovich problem,
attain the same minimal value.
  Moreover we prove a structure theorem for $d$-cyclically monotone sets:
neglecting a set of zero $m$-measure they do not contain any branching
structures, that is, they can be written as the disjoint union of the image of
a disjoint family of geodesics.
