Abstract:Theories with N=2 supersymmetry are among our best tools to understand some of the most pressing questions in fundamental physics, such as that of black hole entropy or those related to strongly coupled phenomena in gauge-theories, e.g. the confinement of the strong nuclear force. I will give general introduction to the subject and review the basic properties of these theories and emphasize the crucial role played by a subset of states that satisfy the so called BPS condition. In the second part of the talk I will discuss my recent work on BPS solutions in low energy N=2 gauge theories. I will show how to solve the BPS equations in general and uncover the role played by a mathematical object called the Hessian potential.