Are there any general conditions guaranteeing that the asymptotic cone of a group/graph is "regular" in some sense? E.g. for $\mathbb{Z}^d$ we get $\mathbb{R}^d$ as the asymptotic cone, which is even a manifold, but for general groups we only get a metric space without additional structure. Does knowing that asymptotic cone is regular (e.g. a manifold) imply any properties of the original group?
Drutu has shown that if every asymptotic cone of the finitely generated group $G$ is a proper space, then $G$ has polynomial growth; and hence by Gromov's Theorem, it follows that $G$ is nilpotentbyfinite. It seems to be open whether or not the conclusion holds if just one asymptotic cone of $G$ is proper. 


I would just like to add to the answer by Simon Thomas that if a group is virtually nilpotent, its asymptotic cones are very regular: they have a Lie group structure and their metric is of CarnotCaratheodory type (these metrics are described in the wikipedia article "SubRiemannian manifold"). Also, the asymptotic cones do not depend on the scaling factor. if a group is not virtually nilpotent, its asymptotic cones tend to be VERY large objects. For example, the asymptotic cones of each nonvirtually cyclic hyperbolic group are real trees with valency $2^{\aleph_0}$ at each point (those groups have exponential growth, I have to admit that I know very little about asymptotic cones of groups of intermediate growth). 

