Kottwitz, building on the work of Langlands and Shelstad, gave a stabilization of the elliptic terms on the geometric side of the trace formula. This stabilization depended on the endoscopic transfer (also known as the fundamental lemma), and so stabilization is now available to us due to the work of Ngo.
What about the spectral side of the trace formula? Has there been any progress towards a stablization of the cuspidal terms? In the introduction to the Paris book project, Harris mentions that this problem is still open, but doesn't provide any details.