Many deep learning models have achieved dominant performance on the offline beat tracking task. However, online beat tracking, in which only the past and present input features are available, still remains challenging. In this paper, we propose BEAt tracking Streaming Transformer (BEAST), an online joint beat and downbeat tracking system based on the streaming Transformer. To deal with online scenarios, BEAST applies contextual block processing in the Transformer encoder. Moreover, we adopt relative positional encoding in the attention layer of the streaming Transformer encoder to capture relative timing position which is critically important information in music. Carrying out beat and downbeat experiments on benchmark datasets for a low latency scenario with maximum latency under 50 ms, BEAST achieves an F1-measure of 80.04% in beat and 52.73% in downbeat, which is a substantial improvement of about 5 and 13 percentage points over the state-of-the-art online beat and downbeat tracking model.