This work develops efficient methods for the implementation of Montague Grammar on a computer. It covers both the syntactic and the semantic aspects of that task. Using a simplified but adequate version of Montague Grammar it is shown how to translate from an English fragment to a purely extensional first-order language which can then be made amenable to standard automatic theorem-proving techniques. Translating a sentence of Montague English into the first-order predicate calculus usually proceeds via an intermediate translation in the typed lambda calculus which is then simplified by lambda-reduction to obtain a first-order equivalent. If sufficient sortal structure underlies the type theory for the reduced translation to always be a first-order one then perhaps it should be directly constructed during the syntactic analysis of the sentence so that the lambda-expressions never come into existence and no further processing is necessary. A method is proposed to achieve this involving the unification of meta-logical expressions which flesh out the type symbols of Montague's type theory with first-order schemas. It is then shown how to implement Montague Semantics without using a theorem prover for type theory. Nothing more than a theorem prover for the first-order predicate calculus is required. The first-order system can be used directly without encoding the whole of type theory. It is only necessary to encode a part of second-order logic and this can be done in an efficient, succinct, and readable manner. Furthermore the pseudo-second-order terms need never appear in any translations provided by the parser. They are vital just when higher-order reasoning must be simulated. The foundation of this approach is its five-sorted theory of Montague Semantics. The objects in this theory are entitites, indices, propositions, properties, and quantities. It is a theory which can be expressed in the language of first-order logic by means of axiom schemas and there is a finite second-order axiomatisation which is the basis for the theorem-proving arrangement. It can be viewed as a very constrained set theory.