In this article, we develop the theory of viscosity solutions for second order fully nonlinear parabolic equations, having a L¹ dependence in time, associated with nonlinear Neumann boundary conditions. The main interests of our study are, in one hand, to revisit the L¹ theory for viscosity solutions and, on the other hand, to extend it to the case of geometrical equations and nonlinear Neumann boundary conditions. We first give a suitable definition of viscosity solutions of this problem, then we provide some very useful properties of these solutions. A key contribution is the comparison results, both adapted to the case of Hamilton-Jacobi-Bellman Equations and to the case of geometrical equations.