This chapter provides an introduction to the field of nonlinear differential equations and dynamical systems, covering key definitions, notation, and fundamental theorems such as Gronwall's inequality. It delves into autonomous equations, discussing phase-space, orbits, critical points, linearization, periodic solutions, first integrals, and Liouville's theorem. The chapter also explores critical points in two-dimensional and three-dimensional systems, periodic solutions using Bendixson's criterion and the Poincaré-Bendixson theorem, and stability theory, including equilibrium and periodic solutions. Additionally, it covers linear equations, stability analysis by linearization and the direct method, perturbation theory, the Poincaré-Lindstedt method, averaging techniques, relaxation oscillations, bifurcation theory, chaos, and Hamiltonian systems. The chapter concludes with appendices on the Morse lemma, linear periodic equations with small parameters, trigonometric formulas, and a sketch of Cotton's proof of the stable and unstable manifold theorem.This chapter provides an introduction to the field of nonlinear differential equations and dynamical systems, covering key definitions, notation, and fundamental theorems such as Gronwall's inequality. It delves into autonomous equations, discussing phase-space, orbits, critical points, linearization, periodic solutions, first integrals, and Liouville's theorem. The chapter also explores critical points in two-dimensional and three-dimensional systems, periodic solutions using Bendixson's criterion and the Poincaré-Bendixson theorem, and stability theory, including equilibrium and periodic solutions. Additionally, it covers linear equations, stability analysis by linearization and the direct method, perturbation theory, the Poincaré-Lindstedt method, averaging techniques, relaxation oscillations, bifurcation theory, chaos, and Hamiltonian systems. The chapter concludes with appendices on the Morse lemma, linear periodic equations with small parameters, trigonometric formulas, and a sketch of Cotton's proof of the stable and unstable manifold theorem.