  This work is devoted to review the modern geometric description of the
Lagrangian and Hamiltonian formalisms of the Hamilton--Jacobi theory. The
relation with the "classical" Hamiltonian approach using canonical
transformations is also analyzed. Furthermore, a more general framework for the
theory is also briefly explained. It is also shown how, from this generic
framework, the Lagrangian and Hamiltonian cases of the theory for dynamical
systems are recovered, and how the model can be extended to other types of
physical systems, such as higher-order dynamical systems and (first-order)
classical field theories in their multisymplectic formulation.
