A graph G is H-saturated for a graph H, if G does not contain a copy of H but adding any new edge to G results in such a copy. An H-saturated graph on a given number of vertices always exists and the properties of such graphs, for example their highest density, have been studied intensively.A graph G is H-induced-saturated if G does not have an induced subgraph isomorphic to H, but adding an edge to G from its complement or deleting an edge from G results in an induced copy of H. It is not immediate anymore that H-induced-saturated graphs exist. In fact, Martin and Smith (2012) showed that there is no P4-induced-saturated graph. Behrens et al. (2016) proved that if H belongs to a few simple classes of graphs such as a class of odd cycles of length at least 5, stars of size at least 2, or matchings of size at least 2, then there is an H-induced-saturated graph.This paper addresses the existence question for H-induced-saturated graphs. It is shown that Cartesian products of cliques are H-induced-saturated graphs for H in several infinite families, including large families of trees. A complete characterization of all connected graphs H for which a Cartesian product of two cliques is an H-induced-saturated graph is given. Finally, several results on induced saturation for prime graphs and families of graphs are provided.