What do random polynomials over finite fields look like?

In this paper, we survey old and new results about random univariate polynomials over a finite field double-struck F signq. We are interested in three aspects: (1) the decomposition of a random polynomial in terms of its irreducible factors, (2) the usage of random polynomials in algorithms, and (3) the average-case analysis of algorithms that use polynomials over finite fields