The perturbative expansion of the pressure of QCD is known to order g^6 log(g), however, the resulting series is poorly convergent at phenomenologically relevant temperatures/couplings. I will discuss a method for improving the convergence of the successive approximations to the QCD pressure in a systematic manner which exactly reproduces the perturbative series in the weak-coupling limit. The method relies on folding in information about the correct high-temperature degrees of freedom via the hard-thermal-loop (HTL) resummation scheme. In order to give some background I will also discuss the poor convergence of quantum mechanics for the ground state of an anharmonic oscillator and present results for the three loop HTL-improved pressure of QED. Finally, I will present new results of an HTL-improved calculation of QCD thermodynamics to three-loops and critically discuss how this compares to available lattice data.