Recent technological advances have made a myriad of soft and flexible electronic devices possible. The essential materials behind many of these devices and systems are electrical conductors that are compliant and retain their conductivity at high strain deformation. These so-called "compliant conductors" are the class of materials that enable stretchable and flexible electrodes, interconnects, and other components utilized in soft electronics. Creating conductors with high compliance, conductivity, and transparency is not a trivial matter, since these properties are often mutually exclusive. Furthermore, engineering reliable compliant conductors with a desired set of properties that remain fairly unchanged over long service lifetimes is an additional criterion that merits careful attention. These challenges have been addressed through at least two primary approaches. The first has been to create conducting composites that are intrinsically stretchable, typically by filling elastomers with conductive particles, or by depositing conductive particles on or just beneath the surface of elastomers. The second strategy has been to build conducting structures capable of reversible bending or stretching. In this review, the key research efforts toward the development of compliant conductors, including transparent conductors, are surveyed for application in flexible and highly stretchable electronic and electromechanical devices.