In Hamiltonian systems, periodic waves often correspond to coherent structures: recurrent, robust patterns that persist over time. Notable examples include water waves, periodic sequences of light pulses in nonlinear optical fibers, and soliton trains in Bose-Einstein condensates. To date, nonlinear stability results for periodic standing or traveling waves in Hamiltonian systems have primarily addressed co-periodic perturbations. A longstanding open problem concerns their stability with respect to localized perturbations: a natural setting in many physical applications. Standard stability arguments in Hamiltonian systems, characterizing solutions as constrained minimizers of an appropriate Lagrangian functional built from conserved quantities of the system, break down in this setting, since localized perturbations preclude coercivity on any finite-codimensional constraint space. In this talk, I propose an alternative framework that integrates variational arguments, Floquet-Bloch decomposition, and Duhamel-based estimates with a modulational ansatz. Our approach yields the first orbital stability results for periodic waves in key Hamiltonian models, such as the Korteweg-de Vries equation, with respect to L^2-localized perturbations. This is joint work with Emile Bukieda (Karlsruhe Institute of Technology).