Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2018, Director: Carlos D'Andrea[en] The Quillen-Suslin theorem is usually stated as "Let $P$ be a finitely generated projective module over $k [ x_{1},...,x_{n}]$ . Then P is free". Before being proven independently and in its full generality by Quillen and Suslin in 1976, this question was usually referred to as the "Serre’s Conjecture", and stood as one of the most relevant open problems in algebra and affine algebraic geometry for twenty years. In this memoir we provide in detail all the algebraic tools needed to have a good understanding of the basic mathematics surrounding this theorem and its more elementary proof by Vaserstein, as well as some algorithms related to it.