  We review the various methods which have been employed recently to describe
the thermodynamics of the high temperature quark-gluon plasma using weak
coupling techniques, and we compare their results with those of most recent
lattice gauge calculations. Many of the difficulties encountered with
perturbation theory at finite temperature are in fact not specific to QCD but
are present in any field theory at finite temperature and will be discussed
first in the simple example of the scalar field theory. We discuss the merits
and limitations of various techniques which have been used to go beyond
perturbation theory in the soft sector, such as dimensional reduction, screened
perturbation theory or hard-thermal-loop perturbation theory, and Phi-derivable
approximations. In the last part of the review, we focus on the later, which
lead to a remarkably simple expression for the entropy of the quark-gluon
plasma. When complemented with further, physically motivated, approximations,
this approach reproduces accurately the entropy obtained from lattice gauge
calculations at temperatures above 2.5 T_c, where T_c is the deconfinement
temperature. This calculation thus provides also support to the physical
picture of the quark-gluon plasma as a gas of weakly interacting
quasiparticles.
