  In this paper we first review the development of high order ADER finite
volume and ADER discontinuous Galerkin schemes on fixed and moving meshes,
since their introduction in 1999 by Toro et al. We show the modern variant of
ADER based on a space-time predictor-corrector formulation in the context of
ADER discontinuous Galerkin schemes with a posteriori subcell finite volume
limiter on fixed and moving grids, as well as on space-time adaptive Cartesian
AMR meshes. We then present and discuss the unified symmetric hyperbolic and
thermodynamically compatible (SHTC) formulation of continuum mechanics
developed by Godunov, Peshkov and Romenski (GPR model), which allows to
describe fluid and solid mechanics in one single and unified first order
hyperbolic system. In order to deal with free surface and moving boundary
problems, a simple diffuse interface approach is employed, which is compatible
with Eulerian schemes on fixed grids as well as direct
Arbitrary-Lagrangian-Eulerian methods on moving meshes. We show some examples
of moving boundary problems in fluid and solid mechanics.
