Introduction.—The dominant subject in the analysis of dynamical systems is the stability of the insuing motion. Many fields of interest under current research involve dynamical systems. These range from fields such as astrodynamics, meteorology, and control systems to biology, chemistry, medicine, and economics. In each of these fields dynamical systems can be formulated, and their stability can be examined as special cases of stability problems in dynamics. During recent years the stability concepts of dynamical systems have been advanced, either by modifying old ideas or by creating new stability concepts. These advances permit a deeper penetration into the more profound stability problems.