In this talk, I will shortly present both a Classical and a Quantum Mechanics framework. I will then notice the main incompatibilities between them - in particular the absence of phase-space trajectories at the quantum level - and explore some possible ways to connect one to the other. This will be done with no appeal to artificial principles of correspondence, but rather with a structured Semiclassical Analysis approach, deploying tools like (pseudo)differential operators and Wigner measures. Last, I will sketch what may come to constitute a complete understanding of classical-quantum correspondences.