This talk examines the design and implementation of embedded boundary methods for problems involving multiphysics – specifically, those involving fluid-fluid and fluid-structure interactions – with an emphasis on those problems involving complex, highly nonlinear phenomena. This talk describes a variety of contributions made to embedded boundary methods for problems involving extreme conditions. First, an implicit time-stepping scheme for embedded boundary methods is designed and implemented, showing speedups of 40 times over the comparable explicit counterpart for problems. This framework is validated on problems involving hydrostatic implosions. Then, a second-order accurate embedded boundary method is introduced for fluid-fluid and fluid-structure interaction. Finally, the accuracy and stability of these embedded boundary methods are examined by introducing tools to quantify the stability of embedded boundary methods and techniques are introduced to stabilize embedded boundary methods as necessary.