In this paper we study solitons invariant with respect to the flow generated by a complete parallel vector field in a ambient Riemannian manifold. A special case occurs when the ambient manifold is the Riemannian product (R×P,dt2+g0) and the parallel field is X=∂t. Similarly to what happens in the Euclidean setting, we call them translating solitons. We see that a translating soliton in R×P can be seen as a minimal submanifold for a weighted volume functional. Moreover we show that this kind of solitons appear in a natural way in the context of a monotonicity formula for the mean curvature flow in R×P. When g0 is rotationally invariant and its sectional curvature is non-positive, we are able to characterize all the rotationally invariant translating solitons. Furthermore, we use these families of new examples as barriers to deduce several non-existence results.