This paper presents a unified end-to-end frame-work for both streaming and non-streamingspeech translation. While the training recipes for non-streaming speech translation have been mature, the recipes for streaming speechtranslation are yet to be built. In this work, wefocus on developing a unified model (UniST) which supports streaming and non-streaming ST from the perspective of fundamental components, including training objective, attention mechanism and decoding policy. Experiments on the most popular speech-to-text translation benchmark dataset, MuST-C, show that UniST achieves significant improvement for non-streaming ST, and a better-learned trade-off for BLEU score and latency metrics for streaming ST, compared with end-to-end baselines and the cascaded models. We will make our codes and evaluation tools publicly available.