There are many versions of the Baum-Connes conjecture (the original, coarse, with coefficients, etc.). I would like to know what group theory results are needed in order to prove or disprove one of these conjectures.
This January in New Orleans Paul Baum gave a rather extensive survey of the history and status of Baum/Connes, so I am guessing that he has a historical survey written up, or quasi-written up, since I am not seeing it on his web page), so I would strongly suggest just asking him. Sadly, I don't believe he is a MO participant.
I want to second David Ben-Zvi's recommendation of Nigel Higson's ICM lecture, though there is a bit of progress that has been published since it was written. Fortunately, you needn't venture further than Nigel's webpage to learn more: his paper "Counterexamples to the Baum-Connes Conjecture" with Lafforgue and Skandalis gives some more recent counter-examples to BC with coefficients and coarse BC.
A few other comments...
As far as I know, the ordinary Baum-Connes conjecture is known to be true for more or less every group that anyone can name. The experts (of which I am not one) seem to suspect that the conjecture is not true in general but that finding a counter-example requires new techniques for constructing discrete groups.
There are a few fairly general techniques for proving the conjecture. The first and most conceptually satisfying is the "Dirac / dual Dirac" method which involves using KK-theory to imitate Atiyah's operator theoretic proof of Bott periodicity. As far as I know there is no evidence that this technique is not powerful enough to prove injectivity in general, but there are examples of groups for which BC is true but the Dirac / dual Dirac method cannot prove surjectivity. I suggest Higson and Guentner's "Group C* algebra's and K-theory" for a discussion of these issues - the paper starts more or less from scratch and reads like a short textbook.
The second method is more of a philosophy: BC tends true for groups which act on nice spaces in a minimally nice way. For example, it is known to be true for a-T-menable groups (see "E-theory and KK-theory for groups which act properly and isometrically on Hilbert space" by Higson and Kasparov). One of the frontiers of the conjecture is in the realm of p-adic groups - I think Paul Baum is still quite active here - and one of the basic techniques seems to be to get p-adic groups to act on buildings.
I hope that helps! For more advice, I would really suggest talking directly to Nigel - I don't think BC is quite as central to his interests these days, but I would guess he's as up on the subject as anyone.
Nigel Higson's 1998 ICM address, "The Baum-Connes Conjecture" (available eg. on his website or on the IMU page) is a great survey of the subject (pre-Lafforgue). In general I highly recommend Higson's writings on the subject (relevant papers, books and talk slides are available at the above link).