Isogeny-based cryptography is one of the few branches of public-key cryptography that promises to resist quantum attacks. The security of these cryptosystems relates to the (presumed) hardness of a variety of computational problems: finding paths in large “isogeny graphs”, computing endomorphisms of elliptic curves, or inverting group actions. We analyse how these problems relate to each other, which are equivalent or harder, with the goal to solidify the foundations of isogeny-based cryptography. Particular care is taken to avoid heuristic analysis: the reductions assume at most the generalised Riemann hypothesis.