This book presents a survey of the relatively new research field of gradient inequalities and their applications. The exposition emphasizes the powerful applications of gradient inequalities in studying asymptotic behavior and stability of gradient-like dynamical systems. It explains in-depth how gradient inequalities are established and how they can be used to prove convergence and stability of solutions to gradient-like systems. This book will serve as an introduction for further studies of gradient inequalities and their applications in other fields, such as geometry and computer sciences. This book is written for advanced graduate students, researchers and applied mathematicians interested in dynamical systems and mathematical modeling.