Given a Cayley graph (G,S) such that |S^n| < Cn^D for some large n, we prove that (G,S) is Gromov-Hausdorff "close" to a connected Lie group G. We give sharp bounds on the dimension and the homogeneous dimension of G respectively in terms of |S^n|/|S| and |S^n|. We also provide quantitative bounds on the ``error" term. We give various applications of this result ranging from GH-limits of Cayley graphs to a conjecture of Benjamini and Kosma on isoperimetry in finite Cayley graphs. This is joint work with Matt Tointon.