Is there any general index theorem for manifold with boundary? My understanding is Atiyah-Patodi-Singer 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 Gromov-Lawson)? 
 A: 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: A general setup for an index theorem on manifolds with boundary, has been developed by the French school in non-commutative geometry, see e.g.
http://www.math.univ-toulouse.fr/~monthube/articles/CMcrascorrection.pdf
and the references therein.
A: *

*The APS theorem works for any  Dirac-type operator; see  e.g. the excellent monograph  by Booss-Wojchiecowski on this topic.


*More than four decades  ago, Boutet de Monvel has described a general set-up for dealing  with boundary value  problems that mimicks  the K-theoretic approach to the  index theorem on closed manifolds.   For a  modern presentation  of this point of  view I recommend this  paper by  Melo-Shrohe-Schick  arXiv: 1203.5649   and the references  therein. It  involves some noncommutative  geometry because  the symbols in the Boutet-de-Monvel calculus of elliptic boundary value  problems define elements in the $K$-theory of a noncommutative    $C^*$-algebra. In the case of closed manifolds symbols  of elliptic operators lead to elements in the $K$-theory of a commutative $C^*$-algebra.
