We discuss new work in a very classical field: the study of branched covers of the Riemann sphere. We first recall the classical picture as developed by Hurwitz, including the relationship between branched covers and group-theoretic monodromy data, and the Hurwitz spaces which parametrize branched covers. We then give two new results: a connectedness result, joint with Fu Liu, for certain Hurwitz spaces in the classical setting, and a result which can be viewed as an analogue of the Riemann existence theorem for certain tamely branched covers of the projective line over fields of positive characteristic.