Background: Arithmetical skills are essential to the effective exercise of citizenship in a numerate society. How these skills are acquired, or fail to be acquired, is of great importance not only to individual children but to the organisation of formal education and its role in society. Method: The evidence on the normal and abnormal developmental progression of arithmetical abilities is reviewed; in particular, evidence for arithmetical ability arising from innate specific cognitive skills (innate numerosity) vs. general cognitive abilities (the Piagetian view) is compared. Results: These include evidence from infancy research, neuropsychological studies of developmental dyscalculia, neuroimaging and genetics. The development of arithmetical abilities can be described in terms of the idea of numerosity--the number of objects in a set. Early arithmetic is usually thought of as the effects on numerosity of operations on sets such as set union. The child's concept of numerosity appears to be innate, as infants, even in the first week of life, seem to discriminate visual arrays on the basis of numerosity. Development can be seen in terms of an increasingly sophisticated understanding of numerosity and its implications, and in increasing skill in manipulating numerosities. The impairment in the capacity to learn arithmetic--dyscalculia--can be interpreted in many cases as a deficit in the concept in the child's concept of numerosity. The neuroanatomical bases of arithmetical development and other outstanding issues are discussed. Conclusions: The evidence broadly supports the idea of an innate specific capacity for acquiring arithmetical skills, but the effects of the content of learning, and the timing of learning in the course of development, requires further investigation.