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 Y where X has genus g, under the hypothesis that n: = (f) is sufficiently large and the monodromy group is not Aₙ or Sₙ. This proves a conjecture of Guralnick and several conjectures of Guralnick and Shareshian.