In Algebraic Topology one is interested in studying properties of topological spaces using algebraic methods. Two topological spaces are considered to be “the same” if they are homeomorphic, and appropriate topological invariants may be used in order to distinguish spaces that are non-homeomorphic. The main such invariants are the homology groups and the homotopy groups. Homology is a tool that counts holes and boundary components of a topological manifold; it appears in a very central way in many branches of mathematics, but also in physics, for example in supersymmetric string theory compactifications as pioneered in Candelas and Horowitz (Nucl Phys B 258:46–74, 1985). Homotopy, on the other hand, is also categorizing topological spaces, but is considering paths on these spaces rather than their boundaries. The simplest among the homotopy groups is the first homotopy group, also called the fundamental group of a topological space. This first chapter is a basic, yet detailed introduction to some standard topological notions that will be important for the understanding of the construction of spectral networks. After some basic notions, we introduce the fundamental group and examine some of its basic properties, as well as a number of applications in distinguishing basic topological spaces up to homeomorphism. Wonderful introductory resources for further information on the topics outlined in this chapter is the textbook by Hatcher (Algebraic topology, 2002) (which is freely available online), as well as tom Dieck (Algebraic topology, 2008); Massey (A basic course in algebraic topology, 1991) and Munkres (Elements of algebraic topology, 1984).