  We construct new explicit non-singular metrics that are complete on
non-compact Riemannian 8-manifolds with holonomy Spin(7). One such metric,
which we denote by A_8, is complete and non-singular on R^8. The other complete
metrics are defined on manifolds with the topology of the bundle of chiral
spinors over S^4, and are denoted by B_8^+, B_8^- and B_8. The metrics on B_8^+
and B_8^- occur in families with a non-trivial parameter. The metric on B_8
arises for a limiting value of this parameter, and locally this metric is the
same as the one for A_8. The new Spin(7) 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.
