Despite the enormous growth in computational resources in machines from laptops to supercomputers, it is still imperative to make the best possible use of available compute power.  Adaptive mesh refinement (AMR), as well as other approaches to variable resolution, reduce the computational cost and memory footprint of a simulation relative to a uniform mesh approach.   AMR has not been widely used in Earth System Models, however, most likely due to the difficulty in identifying objective grid refinement criteria that can work well across a wide range of scales and conditions, as well as the fact that AMR can substantially increase the complexity of the algorithm used to evolve the system in time. Despite the many challenges, EESM researchers have expressed interest in exploring AMR for atmospheric and ocean modeling. This presentation will describe several different approaches to AMR, and discuss our ongoing exploration of using AMR for full 3D simulations of regional ocean circulations as well as 1D subgrid models of atmospheric turbulence and clouds