Sigma (Σ) is a cognitive architecture and system whose development is driven by a combination of four desiderata: grand unification, generic cognition, functional elegance, and sufficient efficiency. Work towards these desiderata is guided by the graphical architecture hypothesis, that key to progress on them is combining what has been learned from over three decades’ worth of separate work on cognitive architectures and graphical models. In this article, these four desiderata are motivated and explained, and then combined with the graphical architecture hypothesis to yield a rationale for the development of Sigma. The current state of the cognitive architecture is then introduced in detail, along with the graphical architecture that sits below it and implements it. Progress in extending Sigma beyond these architectures and towards a full cognitive system is then detailed in terms of both a systematic set of higher level cognitive idioms that have been developed and several virtual humans that are built from combinations of these idioms. Sigma as a whole is then analyzed in terms of how well the progress to date satisfies the desiderata. This article thus provides the first full motivation, presentation and analysis of Sigma, along with a diversity of more specific results that have been generated during its development.