Investigations of classes of grammars that are nontransformational and at the same time highly constrained are of interest both linguistically and mathematically. Context-free grammars (CFG) obviously form such a class. CFGs are not adequate (both weakly and strongly) to characterize some aspects of language structure. Thus how much more power beyond CFG is necessary to describe these phenomena is an important question. Based on certain properties of tree adjoining grammars (TAG) an approximate characterization of class of grammars, mildly context-sensitive grammars (MCSG), has been proposed earlier. In this paper, we have described the relationship between several different grammar formalisms, all of which belong to MCSG. In particular, we have shown that head grammars (HG), combinatory categorial grammars (CCG), and linear indexed grammars (LIG) and TAG are all weakly equivalent. These formalisms are all distinct from each other at least in the following aspects: (a) the formal objects and operations in each formalism, (b) the domain of locality over which dependencies are specified, (c) the degree to which recursion and the domain of dependencies are factored, and (d) the linguistic insights that are captured in the formal objects and operations in each formalism. A deeper understanding of this convergence is obtained by comparing these formalisms at the level of the derivation structures in each formalism. We have described a formalism, the linear context-free rewriting system (LCFR), as a first attempt to capture the closeness of the derivation structures of these formalisms. LCFRs thus make the notion of MCSGs more precise. We have shown that LCFRs are equivalent to muticomponent tree adjoining grammars (MCTAGs), and also briefly discussed some variants of TAGs, lexicalized TAGs, feature structure based TAGs, and TAGs in which local domination and linear precedence are factored TAG(LD/LP).