  In this paper, we establish orbital stability results of cnoidal periodic
waves for the cubic nonlinear Klein-Gordon and Schr\"odinger equations. The
spectral analysis for the corresponding linearized operator is established by
using the Floquet theory and a Morse Index Theorem. First, we prove that the
cnoidal waves for the cubic Klein-Gordon equation are orbitally unstable as a
direct application of Grillakis, Shatah and Strauss' theory. The orbital
stability of cnoidal waves for the Schr\"odinger equation is established by
constructing a suitable Lyapunov functional restricted to the associated zero
mean energy space.
