Copulas are models for the dependence in a random vector and allow to build multivariate models with arbitrary one-dimensional margins. Recently, researchers started to apply copulas to statistical learning problems such as regression or classification. We propose a unified framework for the analysis of such approaches by defining the estimators as solutions of copula-based estimating equations. We present general results on their asymptotic behavior and validity of the bootstrap. The conditions are broad enough to cover most regression-type problems as well as parametric and nonparametric estimators of the copula. The versatility of the method is illustrated with numerical examples and a possible extension to missing data problems.