In this talk, we provide a unifying perspective for the spectral stability of traveling waves in Hamiltonian nonlinear dynamical lattices, of which the Fermi-Pasta-Ulam and the Toda lattice are prototypical examples. We connect two different approaches of studying the stability of traveling waves: one is as an eigenvalue problem for a stationary solution in a co-traveling frame, while the other is as a periodic orbit modulo shifts. We also discuss its parallels with the results on the stability of time-periodic solutions and the Floquet Theory. More importantly, we derive an energy-based spectral stability criterion. It states that a sufficient (but not necessary) condition for a change in the wave stability occurs when the functional dependence of the energy (Hamiltonian) of the model on the wave speed changes its monotonicity. We corroborate this conclusion with a series of analytically and numerically tractable examples.