  We construct a canonical Hausdorff complex analytic moduli space of Fano
manifolds with K\"ahler-Ricci solitons. This naturally enlarges the moduli
space of Fano manifolds with K\"ahler-Einstein metrics, which was constructed
by Odaka and Li-Wang-Xu. We discover a moment map picture for K\"ahler-Ricci
solitons, and give complex analytic charts on the topological space consisting
of K\"ahler-Ricci solitons, by studying differential geometric aspects of this
infinite dimensional moment map. Some stacky words and arguments on
Gromov-Hausdorff convergence help to glue them together in the holomorphic
manner.
