The analysis of correlations within pairs of survival times is of interest to many research topics in medicine, such as the correlation of survival-type endpoints of twins, the correlation of times till failure in paired organs, or the correlation of survival time with a surrogate endpoint. The dependence of such times is assumed monotonic and thus quantification by rank correlation coefficients appropriate. The typical censoring of such times requires more involved methods of estimation and inference as have been developed in recent years. The paper focuses on semiparametric approaches, and in particular on the normal copula-based estimation of Spearman correlation coefficients. The copula approach, often presented for a mathematically inclined readership, is reviewed from the viewpoint of an applied statistician. As an alternative to the maximum likelihood methodology for the normal copula approach (NCE) we introduce an iterative multiple imputation (IMI) method which requires only about 0.05% of the computing time of NCE, without sacrificing statistical performance. For IMI, survival probabilities at death or censoring times are first transformed to normal deviates. Then, those deviates that relate to censored times are iteratively augmented, by using conditional multiple imputation, until convergence is obtained for the normal scores rank correlation, which is similar to Spearman's rank correlation. Statistical properties of NCE and IMI are compared by means of a Monte Carlo study and by means of three real data sets, which also give an impression of the typical range of applications, and of their problems. Copyright © 2013 John Wiley & Sons, Ltd.