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Point-like motile topological defects control the universal dynamics of diverse twodimensional active nematics ranging from shaken granular rods to cellular monolayers. A comparable understanding in higher dimensions has yet to emerge. We report the creation of three-dimensional active nematics by dispersing extensile microtubule bundles in a passive colloidal liquid crystal. Light-sheet microscopy reveals the millimeter-scale structure of active nematics with a single bundle resolution and the temporal evolution of the associated nematic director field. The dominant excitations of three-dimensional active nematics are extended charge-neutral disclination loops that undergo complex dynamics and recombination events. These studies introduce a distinct class of non-equilibrium systems whose turbulent-like dynamics arises from the interplay between internally generated active stresses, the chaotic flows and the topological structure of the constituent defects.