  We study the existence and orbital stability/instability of periodic standing
wave solutions for the Klein-Gordon-Schr\"odinger system with Yukawa and cubic
interactions. We prove the existence of periodic waves depending on the
Jacobian elliptic functions. For one hand, the approach used to obtain the
stability results is the classical Grillakis, Shatah and Strauss theory in the
periodic context. On the other hand, to show the instability results we employ
a general criterium introduced by Grillakis, which get orbital instability from
linear instability.
