  The initial-boundary value problem for the inhomogeneous non-cutoff Boltzmann
equation is a challenging open problem. In this paper, we study the stability
and long-time dynamics of the Boltzmann equation near a global Maxwellian
without angular cutoff assumption in a general $C^3$ bounded domain $\Omega$
(including convex and non-convex cases) with physical boundary conditions:
inflow and Maxwell-reflection (including diffuse reflection) boundary
conditions. We obtain the global-in-time existence, which has an exponential
decay rate for the inflow boundary for both hard and soft potentials, and for
the Maxwell-reflection boundary for hard potentials. The crucial method is to
extend the boundary problem in a bounded domain to the whole space without
regular velocity dissipation, followed by an improved De Giorgi iteration with
a large iteration constant and the $L^2$--$L^\infty$ method.
