The Origins Space Telescope is one of four flagship missions under study for the 2020 Astrophysics Decadal Survey. With a 5.9 m cold (4.5 K) telescope deployed from space, Origins promises unprecedented sensitivity in the near-, mid-, and far-infrared, from 2.8 - 588 $mu$m. This mandates the use of ultra-sensitive and stable detectors in all of the Origins instruments. At the present, no known detectors can meet Origins stability requirements in the near- to mid-infrared, or its sensitivity requirements in the far-infrared. In this work, we discuss the applicability of transition-edge sensors, as both calorimeters and bolometers, to meet these requirements, and lay out a path toward improving the present state-of-the-art.