Recent revisions in the sunspot records illustrate the challenges related to obtaining a 400-year-long observational record of past solar-activity changes. Cosmogenic radionuclides offer the possibility of obtaining an alternative and completely independent record of solar variability. Here, we illustrate that these records offer great potential for quantitative solar-activity reconstructions far back into the past, and we provide updated radionuclide-based solar-activity reconstructions for the past 2000 years. However, cosmogenic-radionuclide records are also influenced by processes independent of solar activity, leading to the need for critical assessment and correction for the non-solar influences. Independent of these uncertainties, we show a very good agreement between the revised sunspot records and the 10 Be records from Antarctica and, in particular, the 14 C-based solar-activity reconstructions. This comparison offers the potential of identifying remaining non-solar processes in the radionuclide-based solar-activity reconstructions, but it also helps identifying remaining biases in the recently revised sunspot records.