  We construct new explicit metrics on complete non-compact Riemannian
8-manifolds with holonomy Spin(7). One manifold, which we denote by A_8, is
topologically R^8 and another, which we denote by B_8, is the bundle of chiral
spinors over $S^4$. Unlike the previously-known complete non-compact metric of
Spin(7) holonomy, which was also defined on the bundle of chiral spinors over
S^4, our new metrics are asymptotically locally conical (ALC): near infinity
they approach a circle bundle with fibres of constant length over a cone whose
base is the squashed Einstein metric on CP^3. We construct the
covariantly-constant spinor and calibrating 4-form. We also obtain an
L^2-normalisable harmonic 4-form for the A_8 manifold, and two such 4-forms (of
opposite dualities) for the B_8 manifold. We use the metrics to construct new
supersymmetric brane solutions in M-theory and string theory. In particular, we
construct resolved fractional M2-branes involving the use of the L^2 harmonic
4-forms, and show that for each manifold there is a supersymmetric example. An
intriguing feature of the new A_8 and B_8 Spin(7) metrics is that they are
actually the same local solution, with the two different complete manifolds
corresponding to taking the radial coordinate to be either positive or
negative. We make a comparison with the Taub-NUT and Taub-BOLT metrics, which
by contrast do not have special holonomy. In an appendix we construct the
general solution of our first-order equations for Spin(7) holonomy, and obtain
further regular metrics that are complete on manifolds B^+_8 and B^-_8 similar
to B_8.
