This book provides an introductory overview of the mathematical theory of dynamical systems, suitable for final-year undergraduate students and for master's studies. Building on the fundamental concepts of topological dynamics and ergodic theory in the first two chapters, the third chapter addresses the central concept of entropy in ergodic theory, which has its origins in statistical physics and information theory, quantifying the complexity of a measure-theoretic dynamical system. The fourth chapter is also dedicated to entropy, but this time within the framework of topological dynamics, where entropy represents a quantitative expression of the deformation of a compact metric space through a continuous transformation. The fifth and final chapter offers a brief insight into current developments.