Complex physical phenomena and systems frequently involve multiple components, complex physics or multi-physics, as well as complex or coupled domains and multiple scales.  Such phenomena are usually modeled by systems of coupled partial differential equations, often nonlinear.  One such phenomenon is electroporoelasticity.  After introducing the equations of electroporoelasticity (the equations of poroelasticity coupled to Maxwell's equations) which have applications in geoscience, hydrology, and petroleum exploration, as well as various areas of science and technology, I will describe some recent results (well posedness), the numerical analysis of a finite-element based method for approximating solutions, and some interesting challenges.