I will discuss uniqueness and existence theorems for four-dimensional, non-compact complete Ricci-flat manifolds with a torus symmetry. Natural asymptotic conditions for these spaces (referred to as `gravitational instantons' are asymptotically flat (S^1 X R^3 with the flat metric), asymptotically locally Euclidean (ALE) and asymptotically Taub-NUT. Solutions are characterised by data (rod structure) that encodes the fixed point sets of the torus action. Furthermore, we establish that for every admissible rod structure there exists an instanton that is smooth up to possible conical singularities at the axes of symmetry. This is in sharp contrast to the analogous problem in the Lorentzian setting (stationary and axisymmetric black hole solutions). I will also discuss generalisations to higher dimensions.