  The Fermi-Pasta-Ulam (FPU) chains of particles in \textit{thermal
equilibrium} are studied from both wave-interaction and particle-interaction
points of view. It is shown that, even in a strongly nonlinear regime, the
chain in thermal equilibrium can be effectively described by a system of weakly
interacting \textit{renormalized} nonlinear waves. These waves possess (i) the
Rayleigh-Jeans distribution and (ii) zero correlations between waves, just as
noninteracting free waves would. This renormalization is achieved through a set
of canonical transformations. The renormalized linear dispersion of these
renormalized waves is obtained and shown to be in excellent agreement with
numerical experiments. Moreover, a dynamical interpretation of the
renormalization of the dispersion relation is provided via a self-consistency,
mean-field argument. It turns out that this renormalization arises mainly from
the trivial resonant wave interactions, i.e., interactions with no momentum
exchange. Furthermore, using a multiple time-scale, statistical averaging
method, we show that the interactions of near-resonant waves give rise to the
broadening of the resonance peaks in the frequency spectrum of renormalized
modes. The theoretical prediction for the resonance width for the thermalized
$\beta$-FPU chain is found to be in very good agreement with its numerically
measured value. Moreover, we show that the dynamical scenario for thermalized
$\beta$-FPU chains is spatially highly localized discrete breathers riding
chaotically on spatially extended, renormalized waves. We present numerical
evidence of existence of discrete breathers in thermal equilibrium.
