In this talk I will present details about a moving mesh finite element method for the approximate solution ofpartial differential equations on an evolving bulk domain in two dimensions, coupled to the solution of partial differential equations on the evolving domain boundary. Problems of this type occur frequently in the modeling of eukaryotic cell migration and chemotaxis - for these applications the bulk domain is either the interior or exterior of the cell and the domain boundary is the cell membrane. Fundamental to the success of the method is the robust generation of bulk and surface meshes for the evolving domains. For this purpose we use a moving mesh partial differential equation (MMPDE) approach. The developed method is applied to model problems with known solutions which indicate second-order spatial and temporal accuracy. The method is then applied to a model of the two-way interaction of a migrating cell with an external chemotactic field.