  In this survey paper, we give an overview of our recent works on the study of
the $W$-entropy for the heat equation associated with the Witten Laplacian on
super-Ricci flows and the Langevin deformation on Wasserstein space over
Riemannian manifolds. Inspired by Perelman's seminal work on the entropy
formula for the Ricci flow, we prove the $W$-entropy formula for the heat
equation associated with the Witten Laplacian on $n$-dimensional complete
Riemannian manifolds with the $CD(K, m)$-condition, and the $W$-entropy formula
for the heat equation associated with the time dependent Witten Laplacian on
$n$-dimensional compact manifolds equipped with a $(K, m)$-super Ricci flow,
where $K\in \mathbb{R}$ and $m\in [n, \infty]$. Furthermore, we prove an
analogue of the $W$-entropy formula for the geodesic flow on the Wasserstein
space over Riemannian manifolds. Our result recaptures an important result due
to Lott and Villani on the displacement convexity of the Boltzmann-Shannon
entropy on Riemannian manifolds with non-negative Ricci curvature. To better
understand the similarity between above two $W$-entropy formulas, we introduce
the Langevin deformation of geometric flows on the cotangent bundle over the
Wasserstein space and prove an extension of the $W$-entropy formula for the
Langevin deformation. Finally, we make a discussion on the $W$-entropy for the
Ricci flow from the point of view of statistical mechanics and probability
theory.
