The conductive polymer poly(3,4-ethylenedioxythiophene) (PEDOT), and especially its complex with poly(styrene sulfonate) (PEDOT:PSS), is perhaps the most well-known example of an organic conductor. It is highly conductive, largely transmissive to light, processible in water, and highly flexible. Much recent work on this ubiquitous material has been devoted to increasing its deformability beyond flexibility—a characteristic possessed by any material that is sufficiently thin—toward stretchability—a characteristic which requires engineering of the structure at the molecular- or nanoscale. Stretchability refers to the ability to accommodate large tensile strains without damage, especially if the extension is reversible. Stretchability is the enabling characteristic of a range of applications envisioned for PEDOT in energy and healthcare. These applications include wearable, implantable, and large-area electronic devices. High degrees of mechanical deformability allow intimate contact with biological tissues and solution-processable printing techniques (e.g., roll-to-roll printing). PEDOT:PSS, however, is only stretchable up to around 10%. In this progress report, we highlight the strategies that have been reported to enhance the stretchability of conductive polymers and composites based on PEDOT and PEDOT:PSS. These strategies include blending with plasticizers or polymers, deposition on elastomers, formation of fibers and gels, and the use of intrinsically stretchable scaffolds for the polymerization of PEDOT.