In this paper we show the local and global well-posedness for the periodic Cauchy problem associated with a special class of 1D Boussinesq systems that emerges in the study of the evolution of long water waves with small amplitude in the presence of surface tension.By a variational approach, we establish the existence of periodic travelling waves.We see that those periodic solutions are characterized as critical points of some functional, for which the existence of critical points follows as a consequence of the Arzela-Ascoli Theorem and the fact that the action functional associated is coercive and is (sequentially) weakly lower semi-continuous in an appropriate set.Contents 1. Introduction 241 2. Local periodic well-posedness 243 3. Global periodic well-posedness for a = c 252 4. Existence of periodic travelling waves via the Arzela-Ascoli Theorem 255 References 260