that $(X,T)$ has a specification-like property one can usually prove that the ergodic invariant measures are abundant: they form a dense subset of the simplex of invariant measures endowed with the weak$^\ast$-topology.

Among the subjects omitted here are: the role of specification-like properties in the theory of large deviations, specification for flows (actions of reals), and other group actions (for example ${\mathbb{Z}}^d$ actions with $d\ge 1$). We are also sure that our catalog of specification-like properties is far from being complete. We have selected only those properties, which have clear connections with Bowen’s original notion of specification. There are many properties which are fitted only to apply to some very specific examples and their relation with the core of this theory remains unclear. There is also a theory developed by Climenhaga and Thompson and their co-authors which is close in spirit to those notions included here. It certainly deserves attention, but regretfully we have had to leave the comparison of this theory with the specification-like properties presented here to another occasion.

We did try to make this paper accessible to non-specialists, but in some places we had to assume that the reader has some experience with topological dynamics and ergodic 