State-of-the-art machine translation (MT) systems are typically trained to generate the "standard" target language; however, many languages have multiple varieties (regional varieties, dialects, sociolects, non-native varieties) that are different from the standard language. Such varieties are often low-resource, and hence do not benefit from contemporary NLP solutions, MT included. We propose a general framework to rapidly adapt MT systems to generate language varieties that are close to, but different from, the standard target language, using no parallel (source--variety) data. This also includes adaptation of MT systems to low-resource typologically-related target languages. We experiment with adapting an English--Russian MT system to generate Ukrainian and Belarusian, an English--Norwegian Bokm{\aa}l system to generate Nynorsk, and an English--Arabic system to generate four Arabic dialects, obtaining significant improvements over competitive baselines.