  In this paper, nonholonomic gerbes will be naturally derived for manifolds
and vector bundle spaces provided with nonintegrable distributions (in brief,
nonholonomic spaces). An important example of such gerbes is related to
distributions defining nonlinear connection (N-connection) structures. They
geometrically unify and develop the concepts of Riemann-Cartan manifolds and
Lagrange-Finsler spaces. The obstruction to the existence of a spin structure
on nonholonomic spaces is just the second Stiefel-Whitney class, defined by the
cocycle associated to a $\mathbb{Z} /2$ gerbe, which is called the nonholonomic
spin gerbe. The nonholonomic gerbes are canonically endowed with N-connection,
Sasaki type metric, canonical linear connection connection and (for odd
dimension spaces) almost complex structures. The study of nonholonomic spin
structures and gerbes have both geometric and physical applications. Our aim is
to prove the Atiyah--Singer theorems for such nonholonomic spaces.
