"The approximation of functions depending on parameters can be facilitated by analysing, a priori, the set of these functions when the parameters vary in a certain domain of dependence. In this talk, we shall remind the classical elements of analysis based on various notions of N-width, such as Kolmogorov or Gelfand N-width involving linear or nonlinear encoders and decoders. We shall also present some new results related to the use of these concepts coupled with neural networks for the decoder that are used in their application to the approximation of the solution of parameter-dependent partial differential equations (PDEs) in an approach that we named ""non-linear reduced basis methods"". We shall also discuss some effective numerical implementations for solving the associated discrete systems and results on some complex problems. The results were obtained in collaboration with Hassan Ballout, Joshua Barnett, Albert Cohen, Charbel Farhat, Christophe Prud'homme and Agustin Somacal."