Various models of mathematical physics admit families of periodic travelling wave solutions. For Hamiltonian PDEs we can characterize coperiodic and modulational stability of those waves by means of analytic criteria relying on an action integral. Since this integral is not explicit it is difficult to check analytically those criteria. The talk will show how the asymptotic analysis of the small amplitude and the long wavelength regimes is a useful complement to numerical simulations in establishing the stability of periodic waves. In particular, a striking connection will be made with the stability of solitary waves.