The operation of gluing manifolds along their boundaries plays a basic role in all of geometric topology. It is used in existence proofs for hyperbolic structures on 3-manifolds (for example in Thurston's work on his geometrization conjecture) and there is much interest in obtaining quantitative control of such operations. That is, given two known manifolds and an identification of their boundaries that yields a hyperbolic manifold, give good estimates of its geometric features. I will sketch aspects of this question throughout the history of the field, including Dehn surgery, Thurston's Skinning Map and the combinatorial geometry of mapping class groups.