We consider Hamilton-Jacobi partial-integro dierential equations (PIDEs in short), where the nonlocal terms involve singular integrals with a Levy type measure of integration. This nonlocal terms naturally arise when studying the innitesimal generator of discontinuous Levy processes. In order to treat the singular terms in a fully nonlinear framework, it is necessary to use a viscosity solution theory in a suitably adapted form. First we review some basic features of the theory, then we focus on some recents results concerning Neumann- boundary value problems for these PIDEs. In particular, we show the occurence of new nonlocal phenomena not encoutered in the case of continuous processes and PDEs. We adopt an analytical approach, i.e. we work directly with the generator and not with the process itself.