We have used high resolution stimulated Brillouin gain spectroscopy to probe the dynamics of glycerol over the temperature ranges 146 to 305 K and 401 to 534 K, which include both the supercooled liquid and glass regimes. The high resolution and large spectral range of the technique have allowed us to resolve Brillouin peaks at low temperatures (146 K) with widths as narrow as 13 MHz and shifts as large as 17.3 GHz. A comparison of the observed Brillouin shifts and linewidths with predictions based on previous work at lower frequencies indicates that the main dispersion in our data arises from the primary (alpha) structural relaxation processes. However, this comparison also reveals that additional relaxation processes, perhaps associated with the secondary (beta) processes, contribute to the Brillouin linewidths both above and below the glass transition. Our results also show a distinct kink in the temperature dependence of the speed of sound at 187 K, the glass transition temperature.