A survey on the Kovalevskaya exponents and their applications

The Kovalevskaya exponents are widely used in the theory of the integrability of dynamical systems. There are many works about this subject. In this survey we list the main results on the Kovalevskaya exponents in integrability theory and nonlinear dynamics. The survey is divided into seven sections. Each section contains a different topic where the Kovalevskaya exponents play a role. For each result we give an example to illustrate its application for a better understanding.