  We survey different approaches to study the invariance of the white noise for
the periodic KdV. We mainly discuss the following two methods. First, we
discuss the PDE method, following Bourgain \cite{BO4}, in a general framework.
Then, we show how it can be applied to the low regularity setting of the white
noise for KdV by introducing the Besov-type space \hat{b}^s_{p, \infty}, sp<
-1. Secondly, we describe the probabilistic method by Quastel, Valk\'o, and the
author \cite{OQV}. We also use this probabilistic approach to study the white
noise for mKdV.
