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It is amazing to see the progress, which has been made over the last two decades, in investigating and developing numerical methods beyond the classical finite element method. The Extended Finite Element Method (XFEM) and related methods are some of the major outcomes of this time. The name ‘XFEM’ has been coined about 10 years ago. Since then, the method has gained the attention and overwhelming interest of an ever-increasing number of researchers. The XFEM is on its best way toward being a reliable and accepted innovative tool for computational engineers in research and industry.

The XFEM proves highly useful for the simulation of solutions that involve non-smooth features such as jumps, kinks, and singularities. This is the case for a large number of engineering applications ranging from crack propagation, two-phase flows, fluid–structure interaction, or even biomechanics. The XFEM has also matured in terms of methodological improvements: different formulations of the approximation, the extension to h- and p-adaptivity, numerical integration, and the treatment of interfacial constraints are some of the topics. Many developments in the closely related Partition of Unity Method and Generalized Finite Element Method have also proven their applicability and usefulness in the context of the XFEM.

Most of the contributions in this special issue are related to presentations held at the first ECCOMAS thematic conference ‘The Extended Finite Element Method—Recent Developments and Applications’ at the RWTH Aachen University, Germany. This event took place in September 2009 and attracted more than 130 participants. The follow-up event will take place in July 2011 in Cardiff, U.K. The editors would like to take the opportunity to thank all authors for their contributions. We wish the reader an interesting and fruitful study of this special issue on the XFEM and related methods.