This paper deals with realization theory of so-called Nash systems, i.e., nonlinear systems the right-hand sides of which are defined by Nash functions. A Nash function is a semialgebraic analytic function. The class of Nash systems is an extension of the class of polynomial and rational systems and it is a subclass of analytic nonlinear systems. Nash systems occur in many applications, including systems biology. Formulation of the realization problem for Nash systems and a partial solution to it are presented. More precisely, necessary and sufficient conditions for realizability of a response map by a Nash system are provided. The concepts of semialgebraic observability and semialgebraic reachability are formulated and their relationship with minimality is explained. In addition to their importance for systems theory, the obtained results are expected to contribute to system identification and model reduction of Nash systems.