The course aims to explain the foundations of nonlinear dynamical systems in relation to the solution of nonlinear differential equations describing physical systems of relevance for engineering applications, and to introduce some of the graphical, mathematical and numerical methods used to study and model nonlinear phenomena. Geometric intuition will aid in keeping the mathematical formalism to a minimum. Real-world examples borrowed from state-of-the-art research in various fields, with a focus on engineering, will aid in attracting the students' interest and in demonstrating the usefulness of the thought methods. In this sense, the "teaching by examples" approach will be adopted.