Much of the activity in linguistics, especially computational linguistics, can be thought of as characterizing not languages simpliciter but relations among languages. Formal systems for characterizing language relations have a long history with two primary branches, based on transducers and synchronous grammars. We present some background on the systems leading to some new results integrating transducers and synchronous grammars through the formal-language-theoretic construct of the bimorphism. We present two applications of synchronous grammars: to tree-adjoining grammar semantics and to syntax-aware machine translation.