Discontinuities arean important domain in the mechanics of solids and fluids. With mechanics focusing on smaller and smaller length scales in order to understand the physics that underly many phenomena that hitherto were modelled in a phenomenological manner, the need to properly model d- continuities increases rapidly. Classical examples are cracks, shear bands and rock faults at a macroscopic level. However, the increase in computational power has made it possible to also analyse phenomena like delamination and debonding in composites (mesoscopic level) and phase boundaries and dis- cation movements at the microscopic and nanoscopic level. While the above examples all pertain to solid mechanics, albeit at a wide range of scales, te- nicallyimportant(moving)?uid-solidinterfacesappearinweldingandcasting processes and in aeroelasticity. Standard discretization methods such as ?nite element, ?nite di?erence or boundaryelementmethods havebeendevelopedfor continuousmedia andare less well suited for treating evolving discontinuities. Indeed, they are appr- imationmethodsforthesolutionofthepartialdi?erentialequations,whichare valid on a domain. Discontinuities divide this domain into two or more parts and at the interface special solution methods must be employed. This holds a fortiori for moving discontinuities such as Luder ¨ s?Piobert bands, Portevin-le- Chatelier bands, solid-state phase boundaries, ?uid-solid interfaces and dis- cations.