Nowadays, neural text generation has made tremendous progress in abstractive summarization tasks. However, most of the existing summarization models take in the whole document all at once, which sometimes cannot meet the needs in practice. Practically, social text streams such as news events and tweets keep growing from time to time, and can only be fed to the summarization system step by step. Hence, in this paper, we propose the task of Stepwise Summarization, which aims to generate a new appended summary each time a new document is proposed. The appended summary should not only summarize the newly added content but also be coherent with the previous summary, to form an up-to-date complete summary. To tackle this challenge, we design an adversarial learning model, named Stepwise Summary Generator (SSG). First, SSG selectively processes the new document under the guidance of the previous summary, obtaining polished document representation. Next, SSG generates the summary considering both the previous summary and the document. Finally, a convolutional-based discriminator is employed to determine whether the newly generated summary is coherent with the previous summary. For the experiment, we extend the traditional two-step update summarization setting to a multi-step stepwise setting, and re-propose a large-scale stepwise summarization dataset based on a public story generation dataset. Extensive experiments on this dataset show that SSG achieves state-of-the-art performance in terms of both automatic metrics and human evaluations. Ablation studies demonstrate the effectiveness of each module in our framework. We also discuss the benefits and limitations of recent large language models on this task.