Copula functions are very convenient for modelling multivariate observations. Popular es- timation methods are the two-stage maximum likelihood and an alternative semi-parametric with empirical cumulative distribution functions (cdf) for the margins. Unfortunately, they can be hastily biased whenever relatively small model deviations occur at the marginal (empirical or parametric) and/or copula levels. In this paper we propose three robust estimators that do not share this undesirable feature. Since heavy skewed and heavy tailed parametric marginals are often considered in applications, we also propose a bounded-bias robust estimator that is corrected for consistency by means of indirect inference. In a simulation study we show that the robust estimators outperform the popular approaches.