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Hurwitz studied branched covers of the Riemann sphere, which can be described equivalently in terms of topology, complex geometry, algebraic geometry, or even -- using fundamental groups and monodromy -- in terms of elementary group theory. The spaces of such covers are now known as Hurwitz spaces, and a fundamental question is to understand their connected components. This question has applications in topology, algebraic geometry and number theory. After reviewing all of the relevant definitions, I will discuss some recent work on the topic, as well as directions for further research.