Constraint-based theories of grammar and grammar formalisms are becoming an increasingly widespread area of research in computational linguistics. Constraint-Based Grammar Formalisms provides the first rigorous mathematical and computational basis for this important area. It introduces new applications to both natural and computer languages and brings together Stuart Shieber's many contributions that have been at the core of developments ranging from the discovery of improved explanations of linguistic phenomena such as binding and coordination to the detailed mathematical analysis of constraint-solving and parsing in a variety of grammar formalisms. This thorough examination of the theoretical and computational foundations of constraint-based grammars and applications to natural-language analysis is unique in several respects. Shieber's theoretical framework may be applied to a whole class of formalisms with properties that make it possible to define a general parsing algorithm for all members of the class, with results that provide essential guidance to the implementer of constraint-based language processing systems. Shieber also brings out new connections between grammatical categories and data types, and between constraint-based natural-language analysis and type inference in computer languages. These connections should be of increasing interest both to computational and theoretical linguists and to computer scientists.