A class of sufficient stability criteria is derived among which the sufficient criteria known as yet are contained as special cases. In this class, there is an optimum criterion. For equilibria periodic along the magnetic field lines (i.e. having some symmetry or closed field lines) its application requires calculating the lowest eigenvalue of a Sturm-Liouville problem. Various more convenient criteria have been obtained from lower bounds of this eigenvalue. These appear well suited for a calculation of lower bounds of a critical β. Some consequences for symmetric equilibria are discussed.