  The categoricity spectrum of a class of structures is the collection of
cardinals in which the class has a single model up to isomorphism. Assuming
that cardinal exponentiation is injective (a weakening of the generalized
continuum hypothesis, GCH), we give a complete list of the possible
categoricity spectrums of an abstract elementary class with amalgamation and
arbitrarily large models. Specifically, the categoricity spectrum is either
empty, an end segment starting below the Hanf number, or a closed interval
consisting of finite successors of the L\"owenheim-Skolem-Tarski number (there
are examples of each type). We also prove (assuming a strengthening of the GCH)
that the categoricity spectrum of an abstract elementary class with no maximal
models is either bounded or contains an end segment. This answers several
longstanding questions around Shelah's categoricity conjecture.
