  In bounded domains, without any geometric conditions, we study the existence
and uniqueness of globally Lipschitz and interior strong C^{1,1}, (and
classical C^2), solutions of general semilinear oblique boundary value problems
for degenerate, (and non-degenerate), augmented Hessian equations, with
strictly regular associated matrix functions. By establishing local second
derivative estimates at the boundary and proving viscosity comparison
principles, we show that the solution is correspondingly smooth near boundary
points where the appropriate uniform convexity is satisfied.
