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The main objective of this work is to study certain types of stability of the origin relatively to one system of Ordinary Differential Equations. We observe that the origin need not to be an equilibrium point of the system under consideration. Such types of stability that may play an important ro le in the theory of adaptive control systems, are known as eventual stability. The concept of eventual stability for systems of or dinary differential equations is a natural extension of the concept of stability in the sense of Lyapunov. We present definitions, basic facts and qualitative properties of the solutions of the menctioned system. We improve results and examples discussed by 9J.P. La Salle, R.J. Rath and V. Lakshmikanthan. The techniques used in this dissertation are provided by the ideas contained in the theory of Lyapunov functions.