The European Conference on Numerical Mathematics and Advanced Applications (ENUMATH) is a series of meetings held every two years to provide a forum for discussion on recent aspects of numerical mathematics and their applications. These proceedings collect the major part of the lectures given at ENUMATH 2005, held in Santiago de Compostela, Spain, from July 18 to 22, 2005. Topics include applications such as fluid dynamics, electromagnetism, structural mechanics, interface problems, waves, finance, heat transfer, unbounded domains, numerical linear algebra, convection-diffusion, as well as methodologies such as a posteriori error estimates, discontinuous Galerkin methods, multiscale methods, optimization, etc.