We compute tautological integrals over Quot schemes on curves and surfaces. After obtaining several explicit formulas over Quot schemes of dimension-0 quotients on curves (and finding a new symmetry), we apply the results to tautological integrals against the virtual fundamental classes of Quot schemes of dimension 0 and 1 quotients on surfaces (using also universality, torus localization and cosection localization). The virtual Euler characteristics of Quot schemes of surfaces, a new theory parallel to the Vafa-Witten Euler characteristics of the moduli of bundles, is defined and studied. Complete formulas for the virtual Euler characteristics are found in the case of dimension-0 quotients on surfaces. Dimension-1 quotients are studied on K3 surfaces and surfaces of general type, with connections to the Kawai-Yoshioka formula and the Seiberg-Witten invariants, respectively. The dimension-1 theory is completely solved for minimal surfaces of general type admitting a nonsingular canonical curve. Along the way, we find a new connection between weighted tree counting and multivariate Fuss-Catalan numbers, which is of independent interest.
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