Existence theorems for generator polynomials over finite fields

Let F_q denote the finite field of prime power order q and F_q^n its extension of degree n. Generator polynomials over F_q of degree n include primitive polynomials, whose roots each generate F_q multiplicatively, and normal polynomials, when the roots form a normal basis { \alpha, \alpha^q, ..., \alpha^q^{n-1} } of F_q^n over F_q. Using character sums and estimates it can be relatively easy to establish asymptotic theorems about the existence of such polynomials with specified properties in the sense that these theorems are valid provided q or n is sufficiently large. The focus of this talk is to describe existence results (and the methods employed to find them) that are complete for all values of $q$ and $n$ for which the relevant question makes sense, with all exceptions listed.