Consider a finite field F=GF(q) and its degree n extension E=GF(q^n). The Primitive Normal Basis Theorem (proved in 1987) guarantees the existence of an element of E which is simultaneously primitive and free over F; in other words, an element which is simutaneously a multiplicative and additive generator for the field E. Subsequently, there has been interest in the existence of primitive free elements with various extra properties; answers to such existence questions have combinatorial applications, as well as being interesting in their own right. In this talk, I will discuss a framework for answering such questions, and present a new result of this kind, the "Strong Primitive Normal Basis Theorem". (This is joint work with S.D.Cohen.)