  In this paper we study the semiclassical limit of the Schr\"odinger equation.
Under mild regularity assumptions on the potential $U$ which include
Born-Oppenheimer potential energy surfaces in molecular dynamics, we establish
asymptotic validity of classical dynamics globally in space and time for
"almost all" initial data, with respect to an appropriate reference measure on
the space of initial data. In order to achieve this goal we prove existence,
uniqueness and stability results for the flow in the space of measures induced
by the continuity equation.
