Ground-penetrating radar (GPR) is a powerful geophysical tool for efficient, high-resolution mapping of the shallow subsurface. Because of physical and economical limitations, a commonly encountered issue is that the corresponding profiles are incomplete in the sense that measurements are desired where they do not exist for data visualization, interpretation, and imaging. Such missing data may result, for example, from regions along the profile where surveying is not possible; from measurements being collected at a regular interval in time but not in space; or from the choice of a large measurement spacing to favor data coverage over quality. Although a number of methods have been proposed for the interpolation of GPR data to tackle this problem, they typically suffer from rather simplistic assumptions that are not satisfied for many GPR datasets. To address these shortcomings, we consider in this article a novel GPR data reconstruction strategy based on multiple-point geostatistics, where missing GPR data are stochastically simulated and conditioned on existing measurements and patterns observed in a representative training image. A key feature in our approach is the consideration of a multivariate image containing both continuous and categorical GPR reflection amplitude data, which helps to guide the simulations toward realistic structures. To demonstrate the power of this single strategy for multiple data reconstruction needs, we show its successful application to a variety of examples in the context of three problems: gap-filling, trace-spacing regularization, and trace densification.