In this chapter, we define the fundamental group of a (pointed) space as the set of path components of the corresponding loop space and study its properties. Initially, we can imagine that maps from the circle \(S^1\) into a topological space X (that map 1 to a fixed point x) always have a ‘generalised mapping degree’. However, it does not take values in \(\mathbb {Z}\) but in the fundamental group \(\pi _1(X,x)\) . We will develop techniques for calculating these fundamental groups by characterising their behaviour with respect to coverings. Subsequently, we illustrate the results in the case of surfaces.