  We obtain a simple explicit expression for the hyper-Kahler Calabi metric on
the co-tangent bundle of CP^{n+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 L^2-normalisable harmonic
4-form in D=8, and an L^2-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 G_2 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^*S^{n+1} and Calabi T^*\CP^{n+1} metrics. In the
latter case we relate the construction to the hyper-Kahler quotient. We then
use the hyper-K\"ahler quotient to give a quaternionic rederivation of the
Calabi metrics.
