We present SpeechMatrix, a large-scale multilingual corpus of speech-to-speech translations mined from real speech of European Parliament recordings. It contains speech alignments in 136 language pairs with a total of 418 thousand hours of speech. To evaluate the quality of this parallel speech, we train bilingual speech-to-speech translation models on mined data only and establish extensive baseline results on Europarl-ST, VoxPopuli and FLEURS test sets. Enabled by the multilinguality of SpeechMatrix, we also explore multilingual speech-to-speech translation, a topic which was addressed by few other works. We also demonstrate that model pre-training and sparse scaling using Mixture-of-Experts bring large gains to translation performance. The mined data and models are freely available.1 Czech (cs), German (de), English (en), Spanish (es), Estonian (et), Finnish (fi), French (fr), Croatian (hr), Hungarian (hu), Italian (it), Lithuanian (lt), Dutch (nl), Polish (pl), Portuguese (pt), Romanian (ro), Slovak (sk) and Slovenian (sl).