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We compute tautological integrals over Quot schemes on curves and surfaces.After obtaining several explicit formulas over Quot schemes of dimension 0quotients on curves (and finding a new symmetry), we apply the results totautological integrals against the virtual fundamental classes of Quot schemesof dimension 0 and 1 quotients on surfaces (using also universality, toruslocalization, and cosection localization). The virtual Euler characteristics ofQuot schemes of surfaces, a new theory parallel to the Vafa-Witten Eulercharacteristics of the moduli of bundles, is defined and studied. Completeformulas for the virtual Euler characteristics are found in the case ofdimension 0 quotients on surfaces. Dimension 1 quotients are studied on K3surfaces and surfaces of general type with connections to the Kawai-Yoshiokaformula and the Seiberg-Witten invariants respectively. The dimension 1 theoryis completely solved for minimal surfaces of general type admitting anonsingular canonical curve. Along the way, we find a new connection betweenweighted tree counting and multivariate Fuss-Catalan numbers which is ofindependent interest.

Quot schemes of curves and surfaces: virtual classes, integrals, Euler characteristics