In this paper, we consider nonparametric copula inference under bivariate censoring. Based on an estimator of the joint cumulative distribution function, we define a discrete and two smooth estimators of the copula. The construction that we propose is valid for a large number of estimators of the distribution function, and therefore for a large number of bivariate censoring frameworks. Under some conditions on the tails of the distributions, the weak convergence of the corresponding copula processes is obtained in $l^{\infty}([0,1]^2).$ We derive the uniform convergence rates of the copula density estimators deduced from our smooth copula estimators. Investigation on the practical behavior of our estimators is done through a simulation study and two real data applications, corresponding to different censoring settings. We use our nonparametric estimators to define a goodness-of-fit procedure for parametric copula models. A new bootstrap scheme is proposed to compute the critical values.
