This article considers the estimation of the parameters of a copula via a simulated method of moments (MM) type approach. This approach is attractive when the likelihood of the copula model is not known in closed form, or when the researcher has a set of dependence measures or other functionals of the copula that are of particular interest. The proposed approach naturally also nests MM and generalized method of moments estimators. Drawing on results for simulation-based estimation and on recent work in empirical copula process theory, we show the consistency and asymptotic normality of the proposed estimator, and obtain a simple test of overidentifying restrictions as a specification test. The results apply to both iid and time series data. We analyze the finite-sample behavior of these estimators in an extensive simulation study. We apply the model to a group of seven financial stock returns and find evidence of statistically significant tail dependence, and mild evidence that the dependence between these assets is stronger in crashes than booms. Supplementary materials for this article are available online. © 2013 American Statistical Association.