Collapsed spaces with Ricci local bounded covering geometry

A complete Riemannian $n$-manifold $M$ is called $\epsilon$-collapsed, if every unit ball in $M$ has a volume less than $\epsilon$ (while often a bound on ‘curvature’ must be imposed to prevent a rescaling of metric). In 1978, Gromov classified ‘almost flat manifolds’ (or the ‘maximally collapsed manifolds’ with sectional curvature bounded in absolute value by one and small diameter) ; a bounded normal covering space of $M$ is diffeomorphic to the quotient of a simply connected nilpotent Lie group modulo a manifold up to a co-compact lattice. This result has been a corner stone in the collapsing theory of Cheeger-Fukaya-Gromov in 90's that there is a nilpotent structure on any $\epsilon$-collapsed manifold with bounded sectional curvature, and this theory has found important applications in Metric Riemannian geometry.
We will survey some recent development in generalizing the collapsing theory to $\epsilon$-collapsed manifolds of Ricci curvature bounded below and the (incomplete) universal cover of every unit ball in $M$ is not collapsed. The study of these collapsed manifolds is partially fuelled by many constructions of collapsed Calabi-Yau metrics using certain underlying singular nilpotent fibrations.