This is a survey article written for a workshop on L-functions and random
matrix theory at the Newton Institute in July, 2004. The goal is to give some
insight into how well-distributed sets of matrices in classical groups arise
from families of $L$-functions in the context of function fields of curves over
finite fields. The exposition is informal and no proofs are given; rather, our
aim is to illustrate what is true by considering key examples.