Title: SimulS2S: End-to-End Simultaneous Speech to Speech Translation. Abstract: Simultaneous speech to speech translation aims to interpret concurrently with the speech in source language, which is of great importance to the real-time understanding of spoken lectures or conversations. Previous methods usually divide this problem into three stages: simultaneous automatic speech recognition (ASR), simultaneous neural machine translation (NMT), and simultaneous text to speech (TTS), which is not end-to-end and suffers from translation delay and error propagation. In this work, we propose SimulS2S, an end-to-end simultaneous speech to speech translation system that directly translates from source-language speech into target-language speech concurrently, which jointly optimizes speech recognition, text translation and speech synthesis in one sequence to sequence model. SimulS2S consists of a speech encoder and a speech decoder both with a speech segmenter and a wait-$k$ strategy for simultaneous translation. Since simultaneous speech to speech translation is challenging, we propose several key techniques to help the training of SimulS2S: 1) a curriculum learning mechanism to train the model gradually from full-sentence translation to simultaneous translation; 2) two auxiliary tasks: ASR and S2T (speech to text translation) that share the same encoder with SimulS2S model to help the training of the encoder; 3) knowledge distillation to transfer the knowledge from the cascaded NMT and TTS models to the SimulS2S model. Experiments on Fisher Spanish-English conversation translation datasets demonstrate that SimulS2S 1) achieves low translation delay and reasonable translation quality compared with full-sentence speech to speech translation (without simultaneous translation), and 2) although performs worse than but close to the accuracy of simultaneous translation with three-stage cascaded models, demonstrating the potential of end-to-end approach for this challenging task.