The study of Jacobians and their compactifications is central to understanding the geometry of algebraic curves, especially in the context of degenerations and moduli. Compactified Jacobians provide essential tools for extending classical theorems and constructions to singular curves, revealing deep connections between algebraic geometry, combinatorics, and tropical geometry. This mini-course will explore these rich interactions. We will begin with the theory of compactified Jacobians for nodal curves. We will then introduce tropical Jacobians, showing how they arise combinatorially from metric graphs. A central focus will be the Abel map for nodal curves and its resolution, highlighting the interplay between algebraic and tropical geometry. Finally, we will discuss the Torelli theorem for nodal curves, addressing the question of reconstructing a curve from its compactified Jacobian. The course aims to provide a bridge between the classical geometric theory of Jacobians and the modern perspectives offered by tropical geometry and degeneration techniques.