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We consider streaming data transmission over a discrete memoryless channel. A new message is given to the encoder at the beginning of each block and the decoder decodes each message sequentially, after a delay of T blocks. In this streaming setup, we study the fundamental interplay between the rate and error probability in the central limit and moderate deviations regimes and show that: 1) in the moderate deviations regime, the moderate deviations constant improves over the block coding or non-streaming setup by a factor of T and 2) in the central limit regime, the second-order coding rate improves by a factor of approximately √T for a wide range of channel parameters. For both the regimes, we propose coding techniques that incorporate a joint encoding of fresh and previous messages. In particular, for the central limit regime, we propose a coding technique with truncated memory to ensure that a …