My understanding is AtiyahPatodiSinger solved the index theorem for manifold with boundary only for certain types of Dirac operators, correct? There is still no (or no hope to get) uniform theorem for the Dirac operator associated with any Dirac bundle (in the sense of GromovLawson)?




I don't know such a general theorem. The problem is the appropriate formulation of a generalized pseudodifferential elliptic boundary value problem. They have a product metric and a natural notion of boundary of Dirac operator. So the A.P.S condition can be set up. I suggest you to take a look to the corresponding $L^2$ problem on the manifold with a cylinder attached. In that case the A.P.S. formula has been extended by Richard Melrose to the so called $b$metrics. These are metrics that are product type only at the infinity. In general the $K$ theory of the boundary contains obstructions to set up LOCAL elliptic boundary value problems. 


A general setup for an index theorem on manifolds with boundary, has been developed by the French school in noncommutative geometry, see e.g. http://www.math.univtoulouse.fr/~monthube/articles/CMcrascorrection.pdf and the references therein. 

