In this talk, we will first review some structures appearing in the description of mechanical systems at three different (“descending") levels: quantum, semi-classical and classical. We will then review how star products at the quantum level give rise to Poisson structures and formal symplectic groupoids at the classical and semi-classical levels, respectively. In the main part, we discuss Lie-theoretic results related to the “lift” of classical to semi-classical geometry, which can be described entirely by flowing ODEs, and their relation to (Kontsevich’s) direct quantization from the classical into the quantum level. We end by discussing the remaining step from semi-classical into the quantum.