Four different kinds of grammars that can define crossing dependencies in human language are compared here: (1) "context sensitive rewrite" grammars with rules that depend on context; (2) "matching" grammars with constraints that filter the generative structure of the language; (3) "copying" grammars which can copy structures of unbounded size; and (4) "generating" grammars in which crossing dependencies are generated from a finite lexical basis. Context sensitive rewrite grammars are syntactically, semantically and computationally unattractive. Generating grammars have a collection of nice properties that ensure they define only ''mildly context sensitive'' languages, and Joshi has proposed that human languages have those properties too. But for certain distinctive kinds of crossing dependencies in human languages, copying or matching analyses predominate. Some results relevant to the viability of mildly context sensitive analyses and some open questions are reviewed.