This is the proceedings from the ENUMATH 2009 conference in Uppsala, Sweden, in June 29- July 3, 2009, with about 100 papers by the invited speakers and the speakers in the minisymposia and contributed sessions. The volume gives an overview of contemporary techniques, algorithms and results in numerical mathematics, scientific computing and their applications. Examples of methods are finite element methods, multiscale methods, numerical linear algebra, and high performance computing algorithms applied to problems in fluid flow, materials, electromagnetics, and chemistry.