We consider a perturbed Hamiltonian system of n degrees of freedom of the form H=H₀+ε*H₁, where H₀ is integrable and nondegenerate and ε is a small parameter. Let <H₁> be the average value of H₁ along the periodic orbits of a resonant torus of H₀. We review on some older and recent results concerning on the significance of <H₁> on the continuation of the non-isolated periodic orbits of H₀ with respect to ε, the nonexistence of analytic integrals for the perturbed system and the structure of the resonance zone for nonzero ε.