The first main objective of this proposal is to determine topological and Floer-theoretic obstructions to the existence of embedded Maslov-zero Lagrangian desingularizations of special Lagrangian (SL) cones. In particular, several such obstructions are expected to arise from the application of Contact homology techniques to the boundary of a neighbourhood of the singular point.The second main objective is to develop an analytic theory of singular, asymptotically conical, SL submanifolds: specifically, it w ill address deformation and gluing issues.These results will rely on a combination of analytic methods previously used both by the proposed researcher and by the scientist in charge, and of gluing techniques developed by the scientist in charge.The above results will allow for a better understanding of which SL singularities can be desingularized, and thus of how to compactify SL moduli spaces.This compactification problem is the main obstacle to the construction of new geometric invariants based on SLs. It is also directly related to applications to Mirror Symmetry and to String Theory.