A unified theory of language combines a Bayesian cognitive linguistic model of language processing, with the proposal that language evolved by sexual selection for the display of intelligence. The theory accounts for the major facts of language, including its speed and expressivity, and data on language diversity, pragmatics, syntax and semantics. The computational element of the theory is based on Construction Grammars. These give an account of the syntax and semantics of the worlds languages, using constructions and unification. Two novel elements are added to construction grammars: an account of language pragmatics, and an account of fast, precise language learning. Constructions are represented in the mind as graph like feature structures. People use slow general inference to understand the first few examples they hear of any construction. After that it is learned as a feature structure, and is rapidly applied by unification. All aspects of language (phonology, syntax, semantics, and pragmatics) are seamlessly computed by fast unification; there is no boundary between semantics and pragmatics. This accounts for the major puzzles of pragmatics, and for detailed pragmatic phenomena. Unification is Bayesian maximum likelihood pattern matching. This gives evolutionary continuity between language processing in the human brain, and Bayesian cognition in animal brains. Language is the basis of our mind reading abilities, our cooperation, self esteem and emotions; the foundations of human culture and society.