Property theory is, as I have understood it, first of all characterized by an attempt to approach naive comprehension type-freely and without committing to extensionality.
There is e.g. the work of Raymond Turner in A Theory of Properties in JSL 52(2) 1987 and his monograph Truth and Modality for Knowledge Representation (Pitman) 1990 as well as Semantics and Property Theory by Gennaro Chierchia and Raymond Turner in Linguistics and Philosophy, Vol. 11, No. 3 (Aug., 1988), pp. 261-302, plus works cited.
Are there any important new developments of property theory beyond the work mentioned?