The exploration of meaning in human languages has traditionally focused on the relation between linguistic forms and what they refer to in the world. Most approaches to formal semantics have been driven by this preoccupation. Recent years, however, have seen the growth of a parallel preoccupation, that of exploring the relation between patterns of meaning and grammatical structure, leading to the search for a restricted subset of meanings that interact with the grammatical system of human languages. As part of this research program of "grammatical semantics", this collection of papers addresses questions of what the atomic elements of grammatically relevant meanings should be, how they must be represented, and how these representations interact with representations of other dimensions of language structure. The papers converge on the idea that morpheme-internal and phrasal meanings are represented along a single dimension, distinct from syntactic representations. Among the phenomena described are alternations of aspectual classes of predicates, event modification and event elaboration, and presuppositions, and the languages include English,Malay, Malayalam, and Mandarin.