A number is normal to the base r if, in its expansion to that base, all possible digit strings of length t are equally frequent for each t. While it is generally believed that many familiar irrational constants are normal, normality has only been proven for numbers expressly invented for the purpose of proving their normality. In this study we give an overview of the main results to date. We then define a new normality criterion, strong normality, to exclude certain normal but clearly non-random artificial numbers. We show that strongly normal numbers are normal but that Champernowne's number, the best-known example of a normal number, fails to be strongly normal. We also re-frame the question of normality as a question about the frequency of mod- , ular residue classes of a sequence of integers. This leads to the beginning of a detailed examination of the digits of square roots.