Euclidean functionals having a certain ``quasi-additivity'' property are shown to provide a general approach to the limit theory of a broad class of random processes which arise in stochastic matching problems. Via the theory of quasi-additive functionals, we obtain a Beardwood-Halton- Hammersley type of limit theorem for the TSP, MST, minimal matching, Steiner tree, and Euclidean semi-matching functionals. One minor but technically useful consequence of the theory is that it often shows that the asymptotic behavior of functionals on the d-dimensional cube coincides with the behavior of the same functional defined on the d-dimensional torus. A more pointed consequence of the theory is that it leads in a natural way to a result on rates of convergence. Finally, we show that the quasi-additive functionals may be approximated by a heuristic with polynomial mean execution time.