We will start by reviewing the classical theory of ordinary homogeneous linear differential equations (over a differential field $K$ of characteristic zero) and Picard-Vessiot extensions. Immediately after we will talk about their Galois group of differential automorphisms, as a linear algebraic group, the natural Galois correspondence, and discuss a few examples over C(t). I will first assume that the constants of $K$ are algebraically closed, then explain the difficulties that arise when we remove this assumption and some recent results around this. In the last part I will briefly explain the several generalizations (and some open questions) to the case of strongly normal extensions and several commuting derivations.