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We prove that the quantum moduli algebra associated to a possibly puncturedcompact oriented surface and a complex semisimple Lie algebra $\mathfrak{g}$ isa Noetherian and finitely generated ring; if the surface has punctures, weprove also that it has no non-trivial zero divisors. Moreover, we show that thequantum moduli algebra is isomorphic to the skein algebra of the surface,defined by means of the Reshetikhin-Turaev functor for the quantum group$U_q(\mathfrak{g})$, and which coincides with the Kauffman bracket skeinalgebra when $\mathfrak{g}=\mathfrak{sl}_2$.

Unrestricted quantum moduli algebras, III: Surfaces of arbitrary genus and skein algebras