Limits of language can be studied from various perspectives such as morphology, syntax, and semantics, among others. At the syntactic level, one direction that has been pursued very actively is via the theory of formal languages, beginning with the so-called Chomsky hierarchy. In this hierarchy, the finite state languages (regular languages) and the context free languages (CFL) have been studied very extensively, leading to many important results relevant to syntax as well as parsing. At the same time, inadequacy of these two classes of grammars (languages) for capturing natural languages has been well documented by now.
A careful look at the pumping lemma for context free languages led to the discovery of the so-called tree adjoining grammars (TAG) and to the notion of mildly context sensitive languages (MCSL), which has served as a framework for showing equivalences among other systems (such as Combinatory Categorial Grammars, CCG and Head Grammars (HG), for example). This has led to a deeper understanding of the limits of language, more specifically, by trying to provide an answer to the question: how far do we need to be beyond CFL to achieve syntactic adequacy.
I will try to describe some of this recent work by a number of researchers in the past few years.

Aravind Joshi did his undergraduate work in India and his graduate work at the University of Pennsylvania in Electrical Engineering, while simultaneously attending some courses in Linguistics at the same University. Since 1961 he has been a faculty member in the Department of Computer and Information Science and the Department of Linguistics. At present, he is a Professor of Computer and Cognitive Science and a Member of the Institute for Research in Cognitive Science.
Besides working on some problems in the mathematics of language, at present, he is involved in a project on discourse annotation, jointly with Bonnie Webber (Edinburgh U.) and Rashmi Prasad (U. Wisconsin), for creating the Penn Discourse Treebank (PDTB).