  In this survey paper, we report on recent works concerning exact
observability (and, by duality, exact controllability) properties of
subelliptic wave and Schr{\"o}dinger-type equations. These results illustrate
the slowdown of propagation in directions transverse to the horizontal
distribution. The proofs combine sub-Riemannian geometry, semi-classical
analysis, spectral theory and non-commutative harmonic analysis.
