  We present a renormalization procedure for Polyakov loops which explicitly
implements the fact that the renormalization constant depends only on the
ultraviolet cutoff. Using this we study the renormalized Polyakov loops in all
representations upto the {\bf 27} of the gauge group SU(3). We find good
evidence for Casimir scaling of the Polyakov loops and for approximate large-N
factorization. By studying many loops together, we are able to show that there
is a matrix model with a single coupling which can describe the high
temperature phase of QCD, although it is hard to construct explicitly. We
present the first results for the non-vanishing renormalized octet loop in the
thermodynamic limit below the SU(3) phase transition, and estimate the
associated string breaking distance and the gluelump binding energy. By
studying the connection of the direct renormalization procedure with a
generalization of an earlier suggestion which goes by the name of the $Q\bar Q$
renormalization procedure, we find that they are functionally equivalent.
