In the lecture I will discuss recent joint work with Richard  Bamler, which uses Ricci flow through singularities to construct deformations of spaces of metrics on 3-manifolds.  We show that the space of metrics with positive scalar on any 3-manifold is either contractible or empty; this extends earlier work by Fernando Marques, which proved path-connectedness.  We also show that for any spherical space form M, the space of metrics with constant sectional curvature is contractible.  This completes the proof of the Generalized Smale Conjecture, and gives a new proof of the original Smale Conjecture for S^3.