  In this paper we initiate a study of first-order rich groups, i.e., groups
where the first-order logic has the same power as the weak second order logic.
Surprisingly, there are quite a lot of finitely generated rich groups, they are
somewhere in between hyperbolic and nilpotent groups (these ones are not rich).
We provide some methods to prove that groups (and other structures) are rich
and describe some of their properties. As corollaries we look at Malcev's
problems in various groups.
