The Goss zeta function is a characteristic-p analogue of the Riemann zeta function for function fields. In the spirit of the Riemann hypothesis, Goss has made several conjectures concerning the distribution of its zeros. We discuss the history of these questions and some recent progress we have made in collaboration with Joe Kramer-Miller. Our main result is a comparison of the distribution of zeros between the higher-genus and genus-zero cases. As a consequence, we are able to prove Goss' conjectures in a large number of previously unknown cases.