Convective transport or the motion of a fluid driven by an unstable density gradient is ubiquitous in nature and the engineering sciences. Of particular significance in the study of this phenomena is the net, time-averaged vertical heat transport induced by the turbulent flow. We will investigate the role that various types of precisely prescribed boundary conditions play in this heat transport using rigorously derived estimates based on the system of partial differential equations described first by Lord Rayleigh in his seminal 1916 paper. These results will be compared to data gathered from state of the art numerical simulations and experimental data. We will also comment on the role of an additional internal heat source, and the effects of rigid body rotation on the heat transport.