Stability of alternative toroidal systems is theoretically studied. It is assumed that, on the one hand, these systems are stabilized by compressibility, and, on the other hand, `according to Mercier', they prove to be unstable. As shown, a remarkable property of `well-organized' alternative systems is the fact that, for such systems, in contrast to `classical' ones (like large-aspect-ratio tokamaks and stellarators), the Mercier modes are the relatively slow sound modes but not the rather fast Alfvén modes. As a result, the ideal instabilities, arising in violation of the Mercier stability criterion, have relatively small growth rates and, in principle, can be suppressed by effects lying beyond the scope of ideal magnetohydrodynamic. In this respect, alternative systems have an advantage over classical ones, for which the violation of the Mercier stability criterion should lead to catastrophic consequences. A new stability criterion, allowing one to distinguish `well-organized' from `badly organized' systems, is derived. To use this criterion one should know the so-called Glasser parameters, calculated by means of averaging certain equilibrium values over the magnetic surface of the system. It is explained that a `candidate' for `well-organized' systems should belong to the class of the finite-aspect-ratio systems. It is noted that, as such `candidates', one can consider, in particular, spherical tokamaks and compact stellarators.