In this talk, we 'll start by reviewing some of the developments on nonlinear dynamical lattices of the discrete nonlinear Schrodinger type. We will explore ideas of continuation from the so-called anti-continuum limit, in order to identify discrete solitons and their stability in 1d lattices, as well as discrete vortices and more complex entities (such as vortex cubes) in two-dimensional and three dimensional case examples. Time-permitting we will present some extensions of these nearest-neighbor lattices to longer range interaction examples and how similar ideas carry over to the setting of Klein-Gordon lattices. More importantly, we 'll venture to go beyond the Hamiltonian realm to a setting which has recently gained significant momentum in the physical community but has very slightly been touched upon in the mathematical literature, namely that of the PT-symmetric lattices. The latter are, in a sense, a very special case example that stands between the Hamiltonian and the dissipative case. We will attempt to illustrate via some prototypical case examples how what we know from the Hamiltonian case is drastically modified in this PT-symmetric setting, highlighting some of the emerging mathematical challenges in this field.