The aim of this project is to study important techniques to determine if two topological spaces are homeomorphic (or homotopy equivalent) or not. Such techniques are the ones related with the fundamental group of a topological space. This project is aimed to cover widely the study of this area of knowledge, always in a very intuitive approach. In chapters one and two there is a motivation of the fundamental group concept. Chapter three shows some important computations throughout some examples. Chapter four introduces the Seifert-Van Kampen theorem and shows its power throughout more examples. This theorem will give some characterisations of really important topological spaces such as topological graphs, connected compact surfaces and torus knots among others. The proof of the Seifert-Van Kampen theorem is given in chapter ve. Finally, chapter six contains a brief analysis of a research paper. In this paper there appear lots of the concepts covered in the project in an applied framework, dierent from the strictly theoric one. This project uses concepts seen in introductory courses of topology and group theory.