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We study the instanton contributions of N=2 supersymmetric gauge theory andpropose that the instanton moduli space is mapped to the moduli space ofpunctured spheres. Due to the recursive structure of the boundary in theDeligne-Knudsen-Mumford stable compactification, this leads to a new recursionrelation for the instanton coefficients, which is bilinear. Instantoncontributions are expressed as integrals on M_{0,n} in the framework of theLiouville F-models. This also suggests considering instanton contributions as akind of Hurwitz numbers and also provides a prediction on the asymptotic formof the Gromov-Witten invariants.  We also interpret this map in terms of the geometric engineering approach tothe gauge theory, namely the topological A-model, as well as in the noncriticalstring theory framework. We speculate on the extension to nontrivialgravitational background and its relation to the uniformization program.Finally we point out an intriguing analogy with the self-dual YM equations forthe gravitational version of SU(2) where surprisingly the same Hauptmodule ofthe SW solution appears.Comment: 58 pages, no figure, harvmac, v2: references added, typos corrected,  v3: references adde

The Liouville Geometry ofN=2 Instantons and the Moduli of Punctured Spheres