In this talk we introduce "self-genericity" axioms. Fixing an ideal I, we define the notion of "M is self-generic" (w.r.t I), where M is an elementary substructure of an initial segment of the universe, and consider several axioms asserting that these structures are frequent: Club Catch, Projective Catch and Stationary Catch (in decreasing order of strength). In particular, we show that Club Catch is equivalent to saturation. We also state some known consistency results related to these axioms, and note some connections with generic embeddings.