Just as we cannot solve every polynomial equation by radicals, many differential equations do not have solutions that could be expressed as elementary functions. Differential Galois Theory is a generalization of the usual Galois theory which studies and explains this phenomenon and helps to understand the relations among solutions of (linear, homogeneous) differential equations. In this series of two introductory talks, I will explain the basic concepts and analogies to usual Galois theory. Moreover, I will mention some recent results and directions of further research.