Growing computing power and investments in the software of Earth system models have increasingly driven impressive improvements in the quality of our global climate and weather simulations. Substantial work is now underway to push these models even further, including new non-hydrostatic formulations, variable resolution grids, exponential and fully implicit time integrators, machine learning methods, and other algorithmic improvements. Nonetheless, questions remain about how these technological improvements have the potential to translate to improving our scientific understanding. In light of this need, we provide a brief review of these technologies in terms of their potential to improve simulation quality and address the outstanding science.