The moving mesh finite element method of Baines, Hubbard and Jimack [1] determines mesh velocities by building the underlying PDE into a monitor conservation principle and recovering an approximation in a manner which preserves the original monitor distribution (so an equidistributed mesh remains so as time progresses). The first part of this talk will outline this method and show some examples which illustrate its ability to track interfaces accurately for implicit moving boundary problems [2]. In theory, it is possible to drive the mesh movement in this algorithm using other monitors, e.g. arc-length, which could be used to distribute mesh nodes more effectively in the interior of the domain. Initial attempts to implement this as a continuous Galerkin finite element method have lacked robustness, so the second part of the talk will outline a discontinuous Galerkin version of the algorithm, capable of stabilising the approximation of the hyperbolic terms introduced when the mesh nodes no longer move according to a physically meaningful conservation principle. Initial results will be shown to demonstrate the feasibility of the approach and the extension to higher orders of accuracy will be discussed. I would particularly like to acknowledge the contributions to this work by Prof Mike Baines (University of Reading), Prof Peter Jimack (University of Leeds) and Mr Tom Radley (University of Nottingham). References: [1] M.J.Baines, M.E.Hubbard and P.K.Jimack, A moving mesh finite element algorithm for the adaptive solution of time-dependent partial differential equations with moving boundaries, Appl. Numer. Math. 54:450--469, 2005. [2] M.J.Baines, M.E.Hubbard and P.K.Jimack, Velocity-based moving mesh methods for nonlinear partial differential equations, Commun. Comput. Phys. 10(3):509--576, 2011.