Ever since the 1970's there have been many efforts to extend the definition of the signature from smooth to singular spaces in a manner that remains invariant under bordisms. The first great success, to spaces with only odd dimensional links, was due to Goresky and MacPherson via the introduction of intersection homology. This was extended to a class of spaces known as `Witt spaces' by Siegel, working topologically, and Cheeger, working analytically. More recently a further extension to `L-spaces' was carried out by Banagl, working topologically, and by A.-Leichtnam-Mazzeo-Piazza, working analytically. I will report on these developments as well as work in progress that extends L-spaces.