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We numerically investigate the morphology and disclination line dynamics of active nematic droplets in three dimensions. Although our model only incorporates the simplest possible form of achiral active stress, active nematic droplets display an unprecedented range of complex morphologies. For extensile activity finger-like protrusions grow at points where disclination lines intersect the droplet surface. For contractile activity, however, the activity field drives cup-shaped droplet invagination, run-and-tumble motion or the formation of surface wrinkles. This diversity of behaviour is explained in terms of an interplay between active anchoring, active flows and the dynamics of the motile dislocation lines. We discuss our findings in the light of biological processes such as morphogenesis, collective cancer invasion and the shape control of biomembranes, suggesting that some biological systems may share the same underlying mechanisms as active nematic droplets.