This book collects the major part of the lectures given at ENUMATH 2005, the Sixth European Conference on Numerical Mathematics and Advanced Applications, that took place in Santiago de Compostela, Spain, from July 18 to 22, 2005. It contents text of invited speakers, and a selection of papers presented in minisymposia and contributed sessions. Topics include applications such as atmosphere and ocean, water pollution, electromagnetism, interface problems, waves, finance, heat transfer, unbounded domains, numerical linear algebra, convection-diffusion, fluid-structure, plates, solids, hyperbolic equations, multiphase flow, Navier-Stokes, singular perturbation problems, non linear PDE, control, parabolic equations, as well as methodologies such as a posteriori error estimates, discontinuous Galerkin methods, multiscale methods, optimization, adaptive methods, domain decomposition techniques, exponential integrators, hp-finite elements, level set methods, fractional step methods, penalty procedures, and finite volumes. The book gives an extensive overview of the most recent research in scientific computing, providing to the reader the latest developments concerning the mathematical issues and the applications of this active field of science.