Optical fibers have often been considered to offer effectively infinite capacity to support the rapid traffic growth essential to our information society. However, as demand has grown and technology has developed, we have begun to realize that there is a fundamental limit to fiber capacity of ~ 100 Tb/s per fiber for systems based on conventional single-core single-mode optical fiber as the transmission medium. This limit arises from the interplay of a number of factors including the Shannon limit, optical fiber nonlinearities, the fiber fuse effect, as well as optical amplifier bandwidth. This article reviews the most recent research efforts around the globe launched over the past few years with a view to overcome these limitations and substantially increase capacity by exploring the last degree of freedom available: the spatial domain. Central to this effort has been the development of brand new fibers for space-division multiplexing and mode-division multiplexing.