We obtain a simple explicit expression for the hyper-Khler Calabi metric on the cotangent bundle of CPn+1, for all n, in which it is constructed as a metric of cohomogeneity one with SU(n+2)/U(n) principal orbits. These results enable us to obtain explicit expressions for an L2-normalisable harmonic 4-form in D=8, and an L2-normalisable harmonic 6-form in D=12. We use the former in order to obtain an explicit resolved M2-brane solution, and we show that this solution is invariant under all three of the supersymmetries associated with the covariantly-constant spinors in the 8-dimensional Calabi metric. We give some discussion of the corresponding dual N=3 three-dimensional field theory. Various other topics are also addressed, including superpotentials for the Calabi metrics and the metrics of exceptional G2 and Spin(7) holonomy in D=7 and D=8. We also present complex and quaternionic conifold constructions, associated with the cone metrics whose resolutions are provided by the Stenzel T*Sn+1 and Calabi T*CPn+1 metrics. In the latter case we relate the construction to the hyper-Khler quotient. We then use the hyper-Khler quotient to give a quaternionic rederivation of the Calabi metrics. 2001 Elsevier Science B.V.