Variational analysis was born and developed in the last quater of the 20th century as a response to the need of optimization and control theories in techniques capable of dealing with non-smoothness and discontinuities of sets, functions and mappings. It was recognized from the outset that efficiency of the new theory in concrete situations heavily depends on how well the behavior and structure of the object is determined by its local approximations provided by the new theory. A search for classes of "good" objects, sufficiently broad to contain practically important types of non-smoothness and discontinuities and, at the same time, void of various pathologies typical for such objects was one of the dominating themes in variational analysis and non-smooth optimization since the very beginning.
Tame sets, functions and mappings that were intensively studied in algebraic geometry approximately at the same time seem to provide an ideal response to this need. In the talk I shall briefly discuss some basic facts of the theory of o-minimal structures and a number of applications demonstrating the power and potential of definable (tame) objects as regards optimization theory.