  We provide a general theory for parallel transport on non-collapsed ${\sf
RCD}$ spaces obtaining both existence and uniqueness results. Our theory covers
the case of geodesics and, more generally, of curves obtained via the flow of
sufficiently regular time dependent vector fields: the price that we pay for
this generality is that we cannot study parallel transport along a single such
curve, but only along almost all of these (in a sense related to the notions of
Sobolev vector calculus and Regular Lagrangian Flow in the nonsmooth setting).
  The class of ${\sf ncRCD}$ spaces contains finite dimensional Alexandrov
spaces with curvature bounded from below, thus our construction provides a way
of speaking about parallel transport in this latter setting alternative to the
one proposed by Petrunin (1998). The precise relation between the two
approaches is yet to be understood.
