Vietoris-Rips complexes of manifolds

I will survey what is known (and mostly unknown) about Vietoris-Rips complexes of manifolds. We might consider a dataset to be nice if it were densely sampled from a manifold. But we do not know what the Vietoris-Rips persistent homology of manifolds looks like, typically! I will focus on Vietoris-Rips complexes of the circle and spheres, comment on ellipsoids and tori, and advertise open questions.