For decades, experiments (and more recently numerical simulations) have attempted to determine how the effective transport of heat (measured by the non-dimensional Nusselt number Nu) scales with the driving force (as measured by the Rayleigh number Ra) --when said driving force is asymptotically strong--in Rayleigh-Benard convection where an incompressible, Boussinesq fluid is driven by an imposed temperature gradient.  To date the results are inconclusive for experiments, and finite limitations on computational resources restrict the potential usefulness of direct numerical simulation.  In contrast to these approaches, rigorous upper bounds on the heat transport have been derived for this system using a variational technique commonly referred to as the background method.  After providing a brief survey of some of the more recent results provided by this method, we show that up to a logarithmic correction, the heat transport is not affected by variations in the thermal boundary condition at infinite Prandtl number.   This is joint work with C. R. Doering and R. Wittenberg.