  Virtual Compton scattering on the proton has been investigated at three yet
unexplored values of the four-momentum transfer $Q^2$: 0.10, 0.20 and 0.45
GeV$^2$, at the Mainz Microtron. Fits performed using either the low-energy
theorem or dispersion relations allowed the extraction of the structure
functions $P_{LL} -P_{TT} / \epsilon$ and $P_{LT}$, as well as the electric and
magnetic generalized polarizabilities $\alpha_{E1}(Q^2)$ and $\beta_{M1}(Q^2)$.
These new results show a smooth and rapid fall-off of $\alpha_{E1}(Q^2)$, in
contrast to previous measurements at $Q^2$ = 0.33 GeV$^2$, and provide for the
first time a precise mapping of $\beta_{M1}(Q^2)$ in the low-$Q^2$ region.
