Let MM be a complete Ricci-flat Kähler manifold with one end and assume that this end converges at an exponential rate to [0,∞)×X[0,∞)×X for some compact connected Ricci-flat manifold XX. We begin by proving general structure theorems for MM; in particular we show that there is no loss of generality in assuming that MM is simply-connected and irreducible with Hol(M)=SU(n)Hol(M)=SU(n), where nn is the complex dimension of MM. If n>2n>2 we then show that there exists a projective orbifold M¯¯¯¯¯M¯ and a divisor D¯¯¯¯∈∣−KM¯¯¯¯¯¯∣D¯∈∣−KM¯∣ with torsion normal bundle such that M¯¯¯¯¯M¯ is biholomorphic to M¯¯¯¯¯∖D¯¯¯¯M¯∖D¯, thereby settling a long-standing question of Yau in the asymptotically cylindrical setting.We give examples where M¯¯¯¯¯M¯ is not smooth: the existence of such examples appears not to have been noticed previously. Conversely, for any such pair M¯¯¯¯¯∖D¯¯¯¯M¯∖D¯ we give a short and self-contained proof of the existence and uniqueness of exponentially asymptotically cylindrical Calabi–Yau metrics on M¯¯¯¯¯∖D¯¯¯¯M¯∖D¯.