Given a Riemannian manifold $M$, when can it be isometrically embedded in Euclidean space? When can a local isometry be found, and when can a global isometry be found? What is the minimum dimension of the target Euclidean space if $M$ has dimension $n$? These questions have been extensively studied during the last century, with perhaps the best known result being the famous Nash embedding theorem from the 1950's. The aim of this talk is to introduce the problem and some of the well known results. The talk is intended to be more of a history lesson rather than technical, so there will be minimal discussion of proofs, and no background required.