Embedded boundary (also known as “cut cell”) methods for solving partial differential equations can significantly reduced cost for grid generation and computation in complex and moving domains. However, they also introduce additional challenges around high accuracy, numerical stability, and other important discretization qualities. I'll present a quick survey of cut cell techniques and some mathematical and software frameworks, along with some examples of model problems and science applications we are developing. Then I will describe our adaptive, higher-order space-time cut cell discretization we have developed for simulating PDE’s on complex, moving domains. Near the domain boundary we use a careful stencil generation procedure that achieves higher-order convergence rates, even when there are boundary kinks or topology changes, or arbitrarily small cells appear. The biggest open questions still involve tradeoffs, including stability (theory of "small cells") and reliability (complex geometry, software) issues, but in some cases these are outweighed by the benefits of conservation, accuracy, and performance. Hans Johansen is a computational researcher at Lawrence Berkeley National Laboratory, specializing in numerical discretizations and algorithms for large-scale scientific computing. He is a participant in the Department of Energy's Exascale Computing Project, focusing on computational science in earth sciences, software technologies, and application performance. He has over 25 years of experience in computational research, IT consulting, startups, and finance services technology. .