This is a joint work with Denis Osin. We show that every group $H$ of at most exponential growth with respect to some left invariant metric admits a bi-Lipschitz embedding into a finitely generated group $G$ such that $G$ is amenable (respectively, solvable, satisfies a non-trivial identity, elementary amenable, of finite decomposition complexity, etc.) whenever $H$ is. We will briefly discuss some applications to subgroup distortion, compression functions of Lipschitz embeddings into uniformly convex Banach spaces, F\o lner functions, and elementary classes of amenable groups.