It is a classical and extremely difficult problem to prove theorems about prime values of irreducible polynomials over the integers. For example, it is still not known if there are infinitely primes of the form $n^2 + 1$. There is a long history of analogies between the integers and polynomials (in one variable) over a finite field, so one can formulate an analogous problem in this other setting. It was discovered several years ago (joint work with K. Conrad and R. Gross) that there are some surprises upon making the translation. After illustrating these surprises via some explicit examples, and discussing theorems that heuristically predict these phenomena, the main goal of the talk is to motivate (by examples) and discuss more recent asymptotic results as the finite field and the polynomial being specialized are allowed to vary. These asymptotic results make precise the sense in which the surprising behavior in the function field case should be rather typical.