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Summary The remap phase in arbitrary Lagrangian–Eulerian (ALE) hydrodynamics involves the transfer of field quantities defined on a post-Lagrangian mesh to some new mesh, usually generated by a mesh optimization algorithm. This problem is often posed in terms of transporting (or advecting) some state variable from the old mesh to the new mesh over a fictitious time interval. It is imperative that this remap process be monotonic, that is, not generate any new extrema in the field variables. It is well known that the only linear methods that are guaranteed to be monotonic for such problems are first-order accurate; however, much work has been performed in developing non-linear methods, which blend both high and low (first) order solutions to achieve monotonicity and preserve high-order accuracy when the field is sufficiently smooth. In this paper, we present a set of methods for enforcing monotonicity targeting high-order discontinuous Galerkin methods for advection equations in the context of high-order curvilinear ALE hydrodynamics. Published 2014. This article is a U.S. Government work and is in the public domain in the USA.