  We establish small-time asymptotic expansions for heat kernels of
hypoelliptic H\"ormander operators in a neighborhood of the diagonal,
generalizing former results obtained in particular by M\'etivier and by Ben
Arous. The coefficients of our expansions are identified in terms of the
nilpotentization of the underlying sub-Riemannian structure. Our approach is
purely analytic and relies in particular on local and global subelliptic
estimates as well as on the local nature of small-time asymptotics of heat
kernels. The fact that our expansions are valid not only along the diagonal but
in an asymptotic neighborhood of the diagonal is the main novelty, useful in
view of deriving Weyl laws for subelliptic Laplacians. In turn, we establish a
number of other results on hypoelliptic heat kernels that are interesting in
themselves, such as Kac's principle of not feeling the boundary, asymptotic
results for singular perturbations of hypoelliptic operators, global smoothing
properties for selfadjoint heat semigroups.
