The classical mechanics, exact quantum mechanics and semiclassical quantum mechanics of the billiard in the triaxial ellopsoid are investigated. The system is separable in ellipsoidal coordinates. A smooth description of the motion is given in terms of a geodesic flow on a solid torus, which is a fourfold cover of the interior of the ellipsoid. Two crossing separatrices lead to four generic types of motion. The action variables of the system are integrals of a single Abelian differential of second kind on a hyperelliptic curve of genus 2. The classical separability carries over to quantum mechanics giving two versions of generalized Lame equations according to the two sets of classical coordinates. The quantum eigenvalues define a lattice when transformed to classical action space. Away from the separatrix surfaces the lattice is given by EBK quantization rules for the four types of classical motion. The transition between the four lattices is described by a uniform semiclassical quantization scheme based on a WKB ansatz. The tunneling between tori is given by penetration integrals which again are integrals of the same Abelian differential that gives the classical action variables. It turns out that the quantum mechanics of ellipsoidal billiards is semiclassically most elegantly explained by the investigation of its hyperelliptic curve and the real and purely imaginary periods of a single Abelian differential.
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