1 Introduction -- 1 Automorphic Distributions and the Weyl Calculus -- 2 he Weyl calculus, the upper half-plane, and automorphic distributions -- 3 Eisenstein distributions, Diracโ{128}{153}s comb and Bezoutโ{128}{153}s distribution -- 4 The structure of automorphic distributions -- 5 The main formula: a heuristic approach -- 2 A Higher-level Weyl Calculus of Operators -- 6 A tamer version of the Weyl calculus: the horocyclic calculus -- 7 The higher-level metaplectic representations -- 8 The radial parts of relativistic wave operators -- 9 The higher-level Weyl calculi -- 10 Can one compose two automorphic operators? -- 11 The sharp product of two power-functions: the Weyl case -- 12 Beyond the symplectic group -- 3 The Sharp Composition of Automorphic Distributions -- 13 The Roelcke-Selberg expansion of functions associated with $$\mathfrak{E}_{{{{\nu }_{1}}}}̂{\sharp }\# \mathfrak{E}_{{\nu 2}}̂{\sharp }$$ the continuous part -- 14 The Roelcke-Selberg expansion of functions associated with $$\mathfrak{E}_{{{{\nu }_{1}}}}̂{\sharp }\# \mathfrak{E}_{{\nu 2}}̂{\sharp }$$ the discrete part -- 15 A proof of the main formula -- 16 Towards the completion of the multiplication table -- 4 Further Perspectives -- 17 Another way to compose Weyl symbols -- 18 Odd automorphic distributions and modular forms of non-zero weight -- 19 New perspectives and problems in quantization theory -- Index of Notation