Optical vortex beams, possessing spatial polarization or phase singularities, have intriguing properties such as the ability to yield super-resolved spots under focussing, and the ability to carry orbital angular momentum that can impart torque to objects. In this review, we discuss the means by which optical fibers, hitherto considered unsuitable for stably supporting optical vortices, may be used to generate and propagate such exotic beams. We discuss the multitude of applications in which a new class of fibers that stably supports vortices may be used, and review recent experiments and demonstration conducted with such fibers.