  This dissertation is devoted to the study of thermodynamics for quantum gauge
theories. The poor convergence of quantum field theory at finite temperature
has been the main obstacle in the practical applications of thermal QCD for
decades. In this dissertation I apply hard-thermal-loop perturbation theory,
which is a gauge-invariant reorganization of the conventional perturbative
expansion for quantum gauge theories to the thermodynamics of QED and
Yang-Mills theory to three-loop order. For the Abelian case, I present a
calculation of the free energy of a hot gas of electrons and photons by
expanding in a power series in $m_D/T$, $m_f/T$ and $e^2$, where $m_D$ and
$m_f$ are the photon and electron thermal masses, respectively, and $e$ is the
coupling constant. I demonstrate that the hard-thermal-loop perturbation
reorganization improves the convergence of the successive approximations to the
QED free energy at large coupling, $e \sim 2$. For the non-Abelian case, I
present a calculation of the free energy of a hot gas of gluons by expanding in
a power series in $m_D/T$ and $g^2$, where $m_D$ is the gluon thermal mass and
$g$ is the coupling constant. I show that at three-loop order hard-thermal-loop
perturbation theory is compatible with lattice results for the pressure, energy
density, and entropy down to temperatures $T \sim 2-3\;T_c$. The results
suggest that HTLpt provides a systematic framework that can be used to
calculate static and dynamic quantities for temperatures relevant at LHC.
