For the Hermitian curveHdefined over the finite field, we give a complete classification of Galois coverings ofHof prime degree. The corresponding quotient curves turn out to be special cases of wider families of curves-covered byHarising from subgroups of the special linear groupSL(2,Fq) or from subgroups in the normaliser of the Singer group of the projective unitary group. Since curves-covered byHare maximal over, our results provide some classification and existence theorems for maximal curves having large genus, as well as several values for the spectrum of the genera of maximal curves. For everyq2, both the upper limit and the second largest genus in the spectrum are known, but the determination of the third largest value is still in progress. A discussion on the “third largest genus problem“ including some new results and a detailed account of current work is given.