In this paper, new methodologies for interpolating rainfall data in individual time intervals (ranging from a day to a year) using Gaussian copulas and unsymmetrical v-copulas, with a variety of treatments of altitude as an exogenous variable, are described. For shorter time aggregations, zeros were treated as censored variables. For each selected time step, the marginal distributions of precipitation amounts were modeled using nonparametric density estimators, while the spatial dependence structures were estimated using a maximum likelihood methodology. The methodology was compared to other common geostatistical interpolators such as ordinary kriging and external drift kriging. Several measures of bias and error structure have been used to assess the efficacy of the methods in a range of comparative split-sampling studies. The data set chosen for the study comprises daily precipitation time series over 41 years in three regions in Germany measured at more than 1200 locations over a large area (126,144 km2). Among the many findings in the paper, the ones that stand out are: (i) correlation between precipitation and topography increases with the length of time interval and is significantly improved by directional smoothing of topography; (ii) the copula methods are superior to kriging methods in terms of quality of interpolation (bias and uncertainty estimation); (iii) the treatment of zeros as censored variables improves interpolation quality for daily and pentad values (monthly and annual data in this region present no dry periods); (iv) the copula methods yield full conditional distributions of estimates at a target point, in an interval, improving substantially on the simple uncertainty estimates derived from kriging; and (v) the Gauss copula in particular performs best overall in terms of computational efficiency, combined with useful error profiles in the interpolations, and of all methods is the most realistic in its error estimates.