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In this survey, we review the classical Hamilton Jacobi theory from ageometric point of view in different geometric backgrounds. We propose aHamilton Jacobi equation for different geometric structures attending to oneparticular characterization: whether they fulfill the Jacobi and Leibnizidentities simultaneously, or if at least they satisfy one of them.  In this regard, we review the case of time dependent and dissipative physicalsystems as systems that fulfill the Jacobi identity but not the Leibnitzidentity. Furthermore, we review the contact evolution Hamilton Jacobi theoryas a split off the regular contact geometry, and that actually satisfies theLeibniz rule instead of Jacobi.  Furthermore, we include a novel result, which is the Hamilton-Jacobi equationfor conformal Hamiltonian vector fields as a generalization of the well knownHamilton Jacobi on a symplectic manifold, that is retrieved in the case of azero conformal factor. The interest of a geometric Hamilton Jacobi equation isthe primordial observation that if a Hamiltonian vector field can be projectedinto a configuration manifold by means of a 1-form dW, then the integral curvesof the projected vector field can be transformed into integral curves of theHamiltonian vector field provided that W is a solution of the Hamilton-Jacobiequation. Geometrically, the solution of the Hamilton Jacobi equation plays therole of a Lagrangian submanifold of a certain bundle. Exploiting these featuresin different geometric scenarios we propose a geometric theory for multiplephysical systems depending on the fundamental identities that their dynamicsatisfies. Different examples are pictured to reflect the results provided,being all of them new, except for one that is reassessment of a previouslyconsidered example.

Reviewing the Geometric Hamilton-Jacobi Theory concerning Jacobi and Leibniz identities