The d-dimensional measurement variety of a graph G is the affine variety of squared edge measurements coming from embeddings of G into complex d-space. The geometry of the measurement variety encodes many rigidity-theoretic properties of G, such as whether it is rigid or globally rigid. In this talk I will give an overview of some recent results on this variety. Among other things, I will talk about the measurement variety of globally rigid graphs, the problem of unlabeled reconstruction, and results about the dual of the measurement variety, which is called the generic stress variety of G.