The course is an introduction to notions of mathematical analysis appearing in the theory of metric spaces with applications in economic theory and/or econometrics. We examine topological notions enabled in general metric spaces. Examples are the notion of convergence of sequences of elements, or the continuity of functions defined between them, finitary notions such as compactness, etc. We are also occupied with non-topological notions, such as uniformities and completeness,and their interplay with the topological ones. In this respect we construct a vocabulary which initially enables us to address the issue of approximation of optimization problems and possibly consider relevant applications. Furthermore the aforementioned construction enables us to state and prove a variety of fixed point theorems. We use them in order to establish existence (and occasionally uniqueness and/or approximability) of solutions of general systems of equations. We apply those notions to problems appearing in dynamic optimization, game theory,etc. The combination of the aforementioned applications enables the unified consideration of both the existence of solutions in problems appearing in economic theory as well as the approximation of those (potentially not easily tractable) solutions with ones that are possibly easier to derive.