We review recent results [10, 9, 8] on time-discrete discontinuous Galerkin (dG) methods for advection-diffusion model problems defined on deformable domains and written on the Arbitrary Lagrangian Eulerian (ALE) framework. ALE formulations deal with PDEs on deformable domains upon extending the domain velocity from the boundary into the bulk with the purpose of keeping mesh regularity. We describe the construction of higher order in time numerical schemes enjoying stability properties

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... nt of the arbitrary extension chosen. Our approach is based on the validity of Reynolds' identity for dG methods which generalize to higher order schemes the Geometric Conservation Law (GCL) condition. Stability, a priori and a posteriori error analyses are briefly discussed and illustrated by insightful numerical experiments.