Effective mathematics education for young children (approximately ages 3 to 5) seems to hold great promise for improving later achievement, particularly in low-SES students who are at risk of inferior education from preschool onwards. Yet there is limited understanding of what preschool and kindergarten mathematics education entails and what is required to implement it effectively. This paper attempts to provide insight into three topics central to understanding and improving early childhood mathematics education in the United States. First, we examine young children's mathematical abilities. Cognitive research shows that young children develop an extensive everyday mathematics and are capable of learning more and deeper mathematics than usually assumed. The second topic is the content and components of early childhood mathematics education. We show that the content of mathematics for young children is wide-ranging (number and operations, shape, space, measurement, and pattern) and sometimes abstract. It involves processes of thinking as well as skills and rote memory. Components of early childhood mathematics education range from play to organized curriculum (several research based programs are now available) and intentional teaching. Third, we consider early childhood educators' readiness to teach mathematics. Unfortunately, the typical situation is that they are poorly trained to teach the subject, are afraid of it, feel it is not important to teach, and typically teach it badly or not at all. Finally, we conclude with policy suggestions. The most urgent need is to improve and support both pre-service and in-service teacher training. [Commentaries from Robert C. Pianta and Deborah Stipek are included. Commentaries are individually referenced.]