Scores of exotic hadrons, particularly tetraquarks and pentaquarks in the heavy-quark sector, have been observed in the past 20 years, and more continue to be discovered to this day. Unlike mesons and baryons, such exotics are not mandated to exist under our current understanding of QCD, meaning that each new discovery presents fresh insights into the expansive possibilities of strong-interaction physics. Even the basic architecture of these multiquark states remains an open question. Here we discuss the merits and deficiencies of their best-known dynamical descriptions, recognizing that the eventual universal model for exotics will almost certainly require the synthesis of more than one fundamental paradigm.