Leading singularities are invariants of multi-loop scattering amplitudes (the full amplitude at tree level) obtained by generalized unitarity. In this talk we show how to construct multi-loop leading singularities on twistor space for maximally super-symmetric Yang-Mills (and gravity). Building on the tree-level twistor-string representation of scattering amplitudes, they can be represented as integrals over a moduli space of nodal curves in twistor space. We discuss how this might arise from a conjectural twistor-string path integral representation for the full loop amplitude. We also show how the construction relates to the Grassmannian representation of leading singularities conjectured by Arkani-Hamed et. al.. This shows firstly that all leading singularities can be represented in the Grassmannian, and secondly that the complexity is limited, in particular we conjecture that there are no new leading singularities at beyond 3p loops for NpMHV amplitudes.