  In this paper, we study the global regularity for regular Monge-Amp\`ere type
equations associated with semilinear Neumann boundary conditions. By
establishing a priori estimates for second order derivatives, the classical
solvability of the Neumann boundary value problem is proved under natural
conditions. The techniques build upon the delicate and intricate treatment of
the standard Monge-Amp\`ere case by Lions, Trudinger and Urbas in 1986 and the
recent barrier constructions and second derivative bounds by Jiang, Trudinger
and Yang for the Dirichlet problem. We also consider more general oblique
boundary value problems in the strictly regular case.
