We consider multiphysics applications from algorithmic and architectural perspectives, where “algorithmic” in- cludes both mathematical analysis and computational complexity and “architectural” includes both software and hard- ware environments. Many diverse multiphysics applications can be reduced, en route to their computational simu- lation, to a common algebraic coupling paradigm. Mathematical analysis of multiphysics coupling in this form is not always practical for realistic applications, but model problems representative of applications discussed herein can provide insight. A variety of software frameworks for multiphysics applications have been constructed and refined within disciplinary communities and executed on leading-edge computer systems. We examine several of these, ex- pose some commonalities among them, and attempt to extrapolate best practices to future systems. From our study, we summarize challenges and forecast opportunities. We also initiate a modest suite of test problems encompassing features present in many applications.