Thanks to their ability to capture complex dependence structures, copulas are frequently used to glue random variables into a joint model with arbitrary one-dimensional margins. More recently, they have been applied to solve statistical learning problems such as regression or classification. Framing such approaches as solutions of estimating equations, we generalize them in a unified framework. We derive consistency, asymptotic normality, and validity of the bootstrap for copula-based Z-estimators. The conditions allow for both continuous and discrete data as well as parametric, nonparametric, and semiparametric estimators of the copula and marginal distributions. The versatility of this methodology is illustrated by several theoretical examples, a simulation study, and an application to financial portfolio allocation.