Emergence as a phenomenon is embedded and expressed in the natural world, and in social systems. Introduced nearly 150 years ago in a philosophical context, it has since been applied in nearly every natural and social science. However all of these uses are not congruent, as the range of emergences in this Special Issue reflect as well; this has limited the accumulation of knowledge about emergence, as well as its development as a discipline. The present paper attempts to bring coherence to emergence, by identifying its core characteristics, its primary expressions, and key principles of emergence. Much of the effort is based on the work of Jeffrey Goldstein, who was one of the first to examine the conceptual, mathematical, and social implications of emergence. The article concludes by showing how a science of emergence can be usefully applied to leadership, and entrepreneurship.