We study doubly-periodic instantons, i.e. instantons on the product of a 1-dimensional complex torus $T$ with a complex line $\u2102$, 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 $T\times {\mathbb{P}}^{1}$. The converse statement is also true, namely a holomorphic bundle on $T\times {\mathbb{P}}^{1}$ which is flat on the torus at infinity, and satisfies a stability condition, comes from a doubly-periodic instanton....