A random real number x has the property that for every integer base b, every digit in the base b representation of x occurs with asymptotic frequency 1/b. That is to say that x is absolutely normal. We will discuss ongoing work about the structure of normality and its generalizations to other forms of uniform distributivity. Methods from harmonic analysis apply to these issues in powerful ways. We will describe some of those applications. Finally, we will discuss a proposal to delineate some limits on those methods and the partial results currently available.