Each grammar formalism specifies a domain of locality, i.e., a domain over which various dependencies (for example, syntactic and semantic) can be specified. It turns out that the various properties of a formalism (syntactic, semantic, computational, statistical, and even psycholinguistic), follow, to a large extent, from the initial specification of the domain of locality. In this talk, I will briefly explore a domain of locality specified by structured objects (trees or acyclic graphs) instead of strings, in the context of some linguistic, computational, statistical and psycholinguistic properties. Such studies provide insights into many aspects of strong generative capacity which is relevant to characterizing structural descriptions. Recently, they have also found some applications to the description of secondary and higher structures of some biological sequences.