Model-theoretic properties of free, projective, and flat $S$-acts

This is the second in a series of articles surveying the body of work on the model theory of $S$-acts over a monoid $S$. The first concentrated on the theory of regular $S$-acts. Here we review the material on model-theoretic properties of free, projective, and (strongly, weakly) flat $S$-acts. We consider questions of axiomatizability, completeness, model completeness, and stability for these classes. Most but not all of the results have already appeared; we remark that the description of those monoids $S$ such that the class of free left $S$-acts is axiomatizable, is new.