  In this survey, we review the classical Hamilton Jacobi theory from a
geometric point of view in different geometric backgrounds. We propose a
Hamilton Jacobi equation for different geometric structures attending to one
particular characterization: whether they fulfill the Jacobi and Leibniz
identities simultaneously, or if at least they satisfy one of them.
  In this regard, we review the case of time dependent and dissipative physical
systems as systems that fulfill the Jacobi identity but not the Leibnitz
identity. Furthermore, we review the contact evolution Hamilton Jacobi theory
as a split off the regular contact geometry, and that actually satisfies the
Leibniz rule instead of Jacobi.
  Furthermore, we include a novel result, which is the Hamilton-Jacobi equation
for conformal Hamiltonian vector fields as a generalization of the well known
Hamilton Jacobi on a symplectic manifold, that is retrieved in the case of a
zero conformal factor. The interest of a geometric Hamilton Jacobi equation is
the primordial observation that if a Hamiltonian vector field can be projected
into a configuration manifold by means of a 1-form dW, then the integral curves
of the projected vector field can be transformed into integral curves of the
Hamiltonian vector field provided that W is a solution of the Hamilton-Jacobi
equation. Geometrically, the solution of the Hamilton Jacobi equation plays the
role of a Lagrangian submanifold of a certain bundle. Exploiting these features
in different geometric scenarios we propose a geometric theory for multiple
physical systems depending on the fundamental identities that their dynamic
satisfies. Different examples are pictured to reflect the results provided,
being all of them new, except for one that is reassessment of a previously
considered example.
