Artificial intelligence engineers have developed artifacts that succeed at tasks generally thought to require great intelligence, but have yet to mechanize the common, everyday abilities of human cognition, including its central feature: understanding. Humans understand the world, their lives, and abstract ideas in ways that other animals and engineering artifacts do not. This book examines the phenomenon of human understanding and offers an approach to understanding understanding itself. The process of understanding requires essential abilities we share with other mammals, but that are not yet available to machines. Further, it requires additional abilities that machines and humans both possess but that are absent in non-human mammals, including other primates. However, it also requires some abilities that are still unique to humans. Important among the ways humans understand is the use of non-verbal means - illustrations, diagrams, paintings, and so forth. These aid understanding by grounding abstractions in familiar perceptual abilities. This book examines the use of diagrams in the process of understanding mathematical concepts as a paradigm case of grounding the understanding one thing in terms of another.