In this thesis, we introduce and assess a new adaptive method for solving non-linear parabolic partial differential equations with fixed or moving boundaries, using a moving mesh with continuous finite elements. The evolution of the mesh within the interior of the spatial domain is based upon conserving the distribution of a chosen monitor function across the domain throughout time, where the initial distribution is based upon the given initial data. For the moving boundary cases, the mesh movement at the boundary is governed by a second monitor function. The method is applied with different monitor functions, to the semilinear heat equation in one space dimension, and the porous medium equation in one and two space dimensions. The effects of optimising initial data for chosen monitors will be considered - in these cases, maintaining the initial distribution amounts to equidistribution. A quantification of the effects of a mesh moving away from an equidistribution are considered here, also the effects of tangling, and then untangling a mesh and restarting.