Let $X$ be a projective algebraic variety, the set of solutions of a system of homogeneous polynomial equations. Several classical notions describe how unconstrained the solutions are, i.e., how close $X$ is to projective space: there are notions of rational, unirational, and stably rational varieties. Over the field of complex numbers, these notions coincide in dimensions one and two, but diverge in higher dimensions. In the last years, many new classes of non stably rational varieties were found, using a specialization technique, introduced by C. Voisin. This method also allowed to prove that the rationality is not a deformation invariant in smooth and projective families of complex varieties, this is a joint work with B. Hassett and Y. Tschinkel. In the first part of my talk I will describe some classical examples, as well as the recent examples obtained by the specialization method. I will give more details on this method in the second part of my talk.