Many current grammar formalisms used in computational linguistics take a unification-based approach that use structures (called feature structures) containing sets of feature-value pairs. In this paper, we describe a unification-based approach to Tree Adjoining Grammars (TAG). The resulting formalism (UTAG) retains the principle of factoring dependencies and recursion that is fundamental to TAGs. We also extend the definition of UTAG to include the lexicalized approach to TAGs (see [Schabes et al., 1988]). We give some linguistic examples using UTAG and informally discuss the descriptive capacity of UTAG, comparing it with other unificationbased formalisms. Finally, based on the linguistic theory underlying TAGs, we propose some stipulations that can be placed on UTAG grammars. In particular, we stipulate that the feature structures associated with the nodes in an elementary tree are bounded ( there is an analogous stipulation in GPSG). Grammars that satisfy these stipulations are equivalent to TAG. Thus, even with these stipulations, UTAGs have more power than CFG-based unification grammars with the same stipulations.