Conservation Laws with an $x$-dependent flux and Hamilton-Jacobi equations with an $x$-dependent Hamiltonian are considered within the same set of assumptions. Uniqueness and stability estimates are obtained only requiring sufficient smoothness of the flux/Hamiltonian. Existence is proved without any convexity assumptions under a mild coercivity hypothesis. The correspondence between the semigroups generated by these equations is fully detailed. With respect to the classical Kružkov approach to Conservation Laws, we relax the definition of solution and avoid any restriction on the growth of the flux. A key role is played by the construction of sufficiently many bounded entropy stationary solutions that provide global bounds in time and space.