State-of-the-art machine translation (MT) systems are typically trained to generate “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.