  This paper is concerned with the construction of special metrics on
non-compact 4-manifolds which arise as resolutions of complex orbifold
singularities. Our study is close in spirit to the construction of the
hyperkaehler gravitational instantons, but we focus on a different class of
singularities. We show that any resolution X of an isolated cyclic quotient
singularity admits a complete scalar-flat Kaehler metric (which is hyperkaehler
if and only if c_1(X)=0), and that if c_1(X)<0 then X also admits a complete
(non-Kaehler) self-dual Einstein metric of negative scalar curvature. In
particular, complete self-dual Einstein metrics are constructed on
simply-connected non-compact 4-manifolds with arbitrary second Betti number.
  Deformations of these self-dual Einstein metrics are also constructed: they
come in families parameterized, roughly speaking, by free functions of one real
variable.
  All the metrics constructed here are toric (that is, the isometry group
contains a 2-torus) and are essentially explicit. The key to the construction
is the remarkable fact that toric self-dual Einstein metrics are given quite
generally in terms of linear partial differential equations on the hyperbolic
plane.
