  We introduce a notion of rainbow saturation and the corresponding rainbow
saturation number. This is the saturation version of the rainbow Tur\'an
numbers whose systematic study was initiated by Keevash, Mubayi, Sudakov, and
Verstra\"ete. 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 K\'asonyi and Tuza. We conclude this paper with related open
questions and conjectures.
