Cornulier conjectured that two completely solvable Lie groups are quasi-isometric (QI) if and only if they are isomorphic. This is a very difficult and very open problem. In this talk I will present some of the structure theory of solvable Lie groups and focus on the importance of sublinear distortions to this theory. I will present recent results that contribute to their QI classification, which are interesting (also) because they are based on a (sublinear) weak form of QI. In addition, I hope to review some of the important work of Cornulier and Tessera on Dehn functions.