  In most results concerning bounds on the heat transport in the
Rayleigh-B\'{e}nard convection problem no-slip boundary conditions for the
velocity field are assumed. Nevertheless it is debatable, whether these
boundary conditions reflect the behavior of the fluid at the boundary. This
problem is important in theoretical fluid mechanics as well as in industrial
applications, as the choice of boundary conditions has effects in the
description of the boundary layers and its properties. In fact, different
boundary conditions may inhibit or enhance heat transport. This review presents
a selection of contributions in the theory of rigorous bounds on Nusselt
number, distinguishing and comparing the results for no-slip, free-slip and
Navier-slip boundary conditions.
