Monodromy groups of indecomposable coverings of bounded genus

For each nonnegative integer $g$, we classify the ramification types and monodromy groups of indecomposable coverings of complex curves $f:X\to Y$ where $X$ has genus $g$, under the hypothesis that $n:=\mathrm{°}(f)$ is sufficiently large and the monodromy group is not $A_n$ or $S_n$. This proves a conjecture of Guralnick and several conjectures of Guralnick and Shareshian.