Accretion disc viscosity: how big is alpha?

We consider observational and theoretical estimates of the accretion disc viscosity parameter $\alpha$. We find that in thin, fully-ionized discs, the best observational evidence suggests a typical range $\alpha \sim 0.1 - 0.4$, whereas the relevant numerical simulations tend to derive estimates for $\alpha$ which are an order of magnitude smaller. We discuss possible reasons for this apparent discrepancy.