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We review our current understanding of the progenitors of both long and short duration gamma-ray bursts (GRBs). Constraints can be derived from multiple directions, and we use three distinct strands; (i) direct observations of GRBs and their host galaxies, (ii) parameters derived from modelling, both via population synthesis and direct numerical simulation and (iii) our understanding of plausible analog progenitor systems observed in the local Universe. From these joint constraints, we describe the likely routes that can drive massive stars to the creation of long GRBs, and our best estimates of the scenarios that can create compact object binaries which will ultimately form short GRBs, as well as the associated rates of both long and short GRBs. We further discuss how different the progenitors may be in the case of black hole engine or millisecond-magnetar models for the production of GRBs, and how central engines may provide a unifying theme between many classes of extremely luminous transient, from luminous and super-luminous supernovae to long and short GRBs.