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Wavelength-division multiplexing (WDM) has been the workhorse of data networks, accommodating exponential traffic growth for two decades. Recently, however, progress in WDM capacity research has markedly slowed down as experiments are closely approaching fundamental Shannon limits of nonlinear fiber transmission. Space-division multiplexing (SDM) is expected to further scale network capacities, using parallel strands of single-mode fiber, uncoupled or coupled cores of multicore fiber, or even individual modes of few-mode fiber in combination with multiple-input-multiple-output (MIMO) digital signal processing. At the beginning of a new era in optical communications, we review initial research in SDM technologies and address some of the key challenges ahead.