The field of my research is part of the qualitative theory of ordinary differential equations. In this research, we focus on the behavior of all solutions of nonautonomous ordinary differential equations. How does the variable coefficients of ODE influence the stability of the solutions? The main purpose of this research is to answer this question. In particular, we deal with the problem on the stability in the Lyapunov sense. For example, we treat asymptotic stability, uniform asymptotic stability and exponential stability. By understanding these essence, we aim to contribute to the development of ODE.