This book discusses the act of understanding in mathematics. The first chapter, "Understanding and Meaning," inquires into the various senses and uses of the word "understanding" in ordinary language and discusses the notion of meaning and relations between understanding and meaning. The second chapter, "Components and Conditions of an Act of Understanding," is central to the book and discusses the act of understanding. Chapter 3, "Processes of Understanding," looks at whole processes of understanding, and the roles therein of explanations and validations, examples, previous knowledge, figurative speech, and activity, both practical and intellectual. The question of evaluation of understanding is dealt with in chapter 4, "Good Understanding." One of the problems raised is concerned with the relativity of any evaluation of understanding. Two important determinants of this relativity, namely the developmental stage of the understanding subject and the culture, are the objects of chapter 5, "Developmental and Cultural Constraints of Understanding." Contains 227 references. (MKR)