  We describe the explicit construction of Yang-Mills instantons on ALE spaces,
following the work of Kronheimer and Nakajima. For multicenter ALE metrics, we
determine the abelian instanton connections which are needed for the
construction in the non-abelian case. We compute the partition function of
Maxwell theories on ALE manifolds and comment on the issue of electromagnetic
duality. We discuss the topological characterization of the instanton bundles
as well as the identification of their moduli spaces. We generalize the 't
Hooft ansatz to SU(2) instantons on ALE spaces and on other hyper-Kahler
manifolds. Specializing to the Eguchi-Hanson gravitational background, we
explicitly solve the ADHM equations for SU(2) gauge bundles with second Chern
class 1/2, 1 and 3/2.
