The installation of seismic-insulating systems is one of the most effective methods of earthquake protection of buildings and structures in highly seismic-prone regions. A sizable number investigations and monographs are devoted to problems of the analysis, design, and practical application of such systems [1,2,3,4]. Currently, however, many important problems of the theory of seismic-insulating systems are under discussion, and design solutions are occasionally incorrect. Examples of this kind of errors are brought to light in [5,6].Systems on kinematic supports can be divided into three types: "locally linearizable," "globally linearizable," and nonlinearizable. These types of supports are described in [6]. The first two types are characterized by increasing restoring force with increasing displacements. For "locally linearizable" supports, the restoring force may be considered linear for small turn angles of the support. For example, V. V. Nazin's supports, spherical, and other supports apply to foundations with this type of supports. "Globally linearizable supports" are characterized by the possibility of blocking their movement under small (in-service) loads. This leads to significant nonlinearity of the support near the equilibrium position, but the restoring force becomes close to linear with increasing turn angle of the support. Yu. I. Bezrukov's supports are, for example, referred to as foundations with this type of supports. Nonlinearizable supports are the most complex from the standpoint of performance under load. For these supports, the restoring force decreases with increasing turn angle. In other words, they are characterized by negative stiffness, and the restoring force is represented in the form F = F 0 − cu, where F 0 is the initial restoring force, c is the stiffness coefficient, and u is the displacement of the system. We will call the indicated supports negative-stiffness supports. Supports of this type are used in foundations designed by Yu. D. Cherepinskii, A. V. Kurzanov, and others. The inaccuracies and errors associated with use of seismic-insulating foundations on kinematic supports with negative stiffness are discussed below. Equations are derived for the vibrations of a building on the foundations under consideration. Impossibility of use of traditional methods of the linear-spectral theory for analysis of their earthquake resistance is demonstrated. It is established that the systems under consideration do not possess a natural vibration period, and may have ambiguous solutions for forced vibrations. Algorithms are proposed for evaluation of the earthquake resistance of buildings on such foundations. The possibility of high efficiency of use of the indicated foundations is noted with correct assignment of their parameters.