TITLE: Dynamical systems, numerical experiments and super-computing AUTHOR: Carles Sim\'o Departament de Matem\`atica Aplicada i An\`alisi, Universitat de Barcelona, Gran Via, 585, 08071 Barcelona (Spain) ABSTRACT: Dynamical systems study the evolution models of natural phenomena and the simplified models which help to understand them. They can be given in deterministic form, either by means of ordinary differential equations, partial differential equations or discrete maps. They are useful in all domains of science and technology. In their study tools from all areas of mathematics are used. But for systems with some degree of complexity it is impossible to produce a fairly complete description of the evolution in the space of states, and its dependence with respect to parameters, without using numerical techniques. They are essential for concrete applications and very useful even for theoretical studies. They can be seen as an experimental part of mathematics. In the last years it has become possible to achieve a generalisation in the systematic use of numerical experiments, due to the availability of large arrays of processors working in parallel with a reduced cost. But the impact of new algorithms has been even larger. This makes feasible to face problems of larger and larger complexity. In this lecture some aspects of the general methodology and several examples are presented.