In the late nineteen-eighties much of our research concerned the application of semigroup-theoretic methods to automata and regular languages, and the connection between computational complexity and this algebraic theory of automata. It was during this period that we became aware of the work of Wolfgang Thomas. Thomas had undertaken the study of concatenation hierarchies of star-free regular languages—a subject close to our hearts— by model-theoretic methods. He showed that the levels of the dot-depth hierarchy corresponded precisely to level of the quantifier alternation hierarchy within first-order logic, and applied Ehrenfeucht-Fräıssé games to prove that the dot-depth hierarchy was strict , a result previously obtained by semigroup-theoretic means [4, 18]. Finite model theory, a subject with which we’d had little prior acquaintance, suddenly appeared as a novel way to think about problems that we had been studying for many years. We were privileged to have been introduced to this field by so distinguished a practitioner as Wolfgang Thomas, and to have then had the opportunity to work together with him. The study of languages defined with modular quantifiers, the subject of the present survey, began with this collaboration.