This joint work with I. Bouw addresses the open questions of which inertia groups and conductors can occur for wildly ramified Galois covers of the projective line branched at exactly one point over an algebraically closed field of characteristic p. We prove existence and nonexistence results for such covers with specified group, inertia group and conductor. In particular, we show that the set of conductors which occurs with group $A_p$ differs from the set occuring with group $PSL_2(p)$. The methods use lifting and reduction techniques to show that the existence of covers with given inertia in characteristic $p$ is closely related to the arithmetic of covers in characteristic zero.