Text detoxification is the task of transferring the style of text from toxic to neutral. While here are approaches yielding promising results in monolingual setup, e.g., (Dale et al., 2021; Hallinan et al., 2022), cross-lingual transfer for this task remains a challenging open problem (Moskovskiy et al., 2022). In this work, we present a large-scale study of strategies for cross-lingual text detoxification -- given a parallel detoxification corpus for one language; the goal is to transfer detoxification ability to another language for which we do not have such a corpus. Moreover, we are the first to explore a new task where text translation and detoxification are performed simultaneously, providing several strong baselines for this task. Finally, we introduce new automatic detoxification evaluation metrics with higher correlations with human judgments than previous benchmarks. We assess the most promising approaches also with manual markup, determining the answer for the best strategy to transfer the knowledge of text detoxification between languages.