Here we consider the development and analysis of algorithms for the numerical solution of PDEs that are particularly suited to take advantage of evolving computing hardware - available commodity clusters, hybrid CPU-GPU systems and multicore desktop machines. Such algorithms consist of three modules: (1) a procedure to step forward in time, (2) the computation of a new spatial mesh as required, and (3) the solution of the (physical) PDE on the newly constructed mesh. Domain decomposition (DD) parallelizes a computation by partitioning the spatial domain into subdomains. The solution on each subdomain is computed by individual processors or cores. With appropriate conditions to transmit solution information between cores, the subdomain solutions can be rapidly combined to give a solution to the original problem. The application of DD methods for the physical PDE, step (3) above, is well established. Here we will consider the application of DD to the PDE based mesh generation problem used in step (2) above. If time permits we will show how to couple the Revisionist Integral Deferred Correction approach (for small scale parallelism in time) with domain decomposition to give a fully parallel space-time method.