We construct examples of complete metrics with holonomy G2 and Spin(7). Specifically, on the product of the 3-sphere with real 4-space, we construct a complete SO(4)-invariant metric with holonomy G2, on the bundle of self-dual 2-forms on the complex projective plane, we construct a complete SU(3)-invariant metric with holonomy G2, and on the positive spin bundle over the 4-sphere, we construct a complete Spin(5)-invariant metric with holonomy Spin(7).
The method is to look among the group invariant metrics for one that has the right holonomy. The point, in each case, is that the group acts with cohomogeneity one and so the problem is reduced to an ODE problem. These ODE turn out to be managable. In many ways, the construction is reminiscent of Calabi's construction of a complete metric with holonomy Sp(n) on the holomorphic cotangent bundle of complex projective n-space.
It is now known, from Dominic Joyce's work (see MR 99j:53065 and the references contained therein), that compact examples exist in both cases.
Recently, physicists S. Gukov and coworkers and C. Page and coworkers (not to mention Atiyah and Witten) have produced more examples of cohomogeneity one metrics with exceptional holonomy. Of course, in each case, it `reduces' to a study of a certain ODE system, which can sometimes be integrated.