Progress in sentence simplification has been hindered by a lack of labeled parallel simplification data, particularly in languages other than English. We introduce MUSS, a Multilingual Unsupervised Sentence Simplification system that does not require labeled simplification data. MUSS uses a novel approach to sentence simplification that trains strong models using sentencelevel paraphrase data instead of proper simplification data. These models leverage unsupervised pretraining and controllable generation mechanisms to flexibly adjust attributes such as length and lexical complexity at inference time. We show that this paraphrase data can be mined in any language from Common Crawl using semantic sentence embeddings, thus removing the need for labeled data. We evaluate our approach on English, French, and Spanish simplification benchmarks and closely match or outperform the previous best supervised results, despite not using any labeled simplification data. We push the state of the art further by incorporating labeled simplification data.