Global existence and uniqueness results are established for mixed initial-boundary value problems for degenerate nonlinear parabolic equations of the type ${\partial {\rm u}}\over{\partial {\rm t}}$ = $\phi$(x,$\nabla$u)$\Delta$u + f(x,u,$\nabla$u) where u = u(x,t) is real-valued, x is in a smooth bounded domain $\Omega$ in $\IR\sp{\rm n}$, and t $\geq$ 0. By 'degenerate' we mean that $\phi$(x,$\xi)$ $>$ 0 for x $\epsilon\Omega$ and $\xi\ \epsilon\ \IR\sp{\rm n}$ but possibly $\phi$(x,$\xi)$ = 0 for x $\epsilon\ \partial\ \Omega$. We also consider a variety of boundary conditions, which can be either linear (e.g. Dirichlet, Neumann, Robin or periodic) or nonlinear, in which case it takes the form $-{\partial{\rm u}\over\partial{\rm n}}$ $\epsilon$ $\beta$(u(x,t)) for x $\epsilon\ \partial\Omega$, t $\geq$ 0. Here $\nu$ is the unit outer normal to $\partial\Omega$ at x and $\beta$ is a maximal monotone graph in $\IR$ x $\IR$ containing (0,0). In particular, both the equation and the boundary condition can be nonlinear