We review recent research on nonlinear optical interactions in optical-fiber nanowires(OFNs) with sub-micron transverse dimensions. Such OFNs, which are fabricated from standard optical fibers, offer numerous beneficial optical and mechanical properties, including strong evanescent fields, high flexibility and configurability, a small mass, and low-loss interconnection to other optical fibers and fiberized components. In particular, the strong confinement of light enables a large enhancement of nonlinear interactions and group-velocity dispersion engineering. The combination of these properties makes OFNs ideal for many nonlinear optical applications, including harmonic generation, Brillouin scattering, four-wave mixing, supercontinuum generation, and optomechanics. With the incorporation of new materials, OFNs should be ideally suited for a host of nonlinear optical interactions and devices and offer great potential in miniature fiber devices for optical telecommunications and optical sensor applications.