We introduce a notion of rainbow saturation and the corresponding rainbow saturation number. This is the saturation version of the rainbow Turán numbers whose systematic study was initiated by Keevash, Mubayi, Sudakov, and Verstraëte. We give examples of graphs for which the rainbow saturation number is bounded away from the ordinary saturation number. This includes all complete graphs $K_n$ for $n\geq 4$, and several bipartite graphs. It is notable that there are non-bipartite graphs for which this is the case, as this does not happen when it comes to the rainbow extremal number versus the traditional extremal number. We also show that saturation numbers are linear for a large class of graphs, providing a partial rainbow analogue of a well known theorem of Kásonyi and Tuza. We conclude this paper with related open questions and conjectures.