  We study doubly-periodic instantons, i.e. instantons on the product of a
1-dimensional complex torus T with a complex line C, with quadratic curvature
decay. We determine the asymptotic behaviour of these instantons, constructing
new asymptotic invariants. We show that the underlying holomorphic bundle
extends to TxP1. The converse statement is also true, namely a holomorphic
bundle on TxP1 which is flat on the torus at infinity, and satisfies a
stability condition, comes from a doubly-periodic instanton. Finally, we study
the hyperkahler geometry of the moduli space of doubly-periodic instantons, and
prove that the Nahm transform previously defined by the second author is a
hyperkahler isometry with the moduli space of certain meromorphic Higgs bundles
on the dual torus.
