Connectivity of Hurwitz spaces for A_5 Sergey Shpectorov Hurwitz spaces are the moduli spaces of meromorphic functions on Riemann surfaces. For a given monodromy group G and ramification type C, the connected component of the corresponding Hurwitz space are in a bijection with the orbits of the braid group B_r on the Nielsen class Ni(G,C). We survey the available results about connectivity of Hurwitz spaces starting from the classical result of Clebsch from 1872, and more recent theorems by Fried and by Liu and Osserman. After that we present our theorem, which is the complete classification of connected components of the Hurwitz spaces with monodromy A_5.