Dimension $4$ is the next horizon for applications of Ricci flow to topology, where the main goal is to understand the topological operations that Ricci flow performs both at singular times, and in its long-term behavior.

 With Alix Deruelle, we explain how Ricci flow develops or resolves orbifold singularities by a notion of stability for orbifold Ricci solitons that we introduce depending only on the curvature at the singular points. We construct ancient and immortal Ricci flows spontaneously forming or desingularizing arbitrarily complicated orbifold singularities by bubbling-off Ricci-flat ALE metrics.

 This unexpectedly predicts that singular spherical and cylindrical orbifolds should not appear as finite-time singularity models. On the other hand, (complex) hyperbolic orbifolds appear as limits of immortal "thick" $4$-dimensional Ricci flows as $t\to+\infty$.