Text summarization is a personalized and customized task, i.e., for one document, users often have different preferences for the summary. As a key aspect of customization in summarization, granularity is used to measure the semantic coverage between summary and source document. Coarse-grained summaries can only contain the most central event in the original text, while fine-grained summaries cover more sub-events and corresponding details. However, previous studies mostly develop systems in the single-granularity scenario. And models that can generate summaries with customizable semantic coverage still remain an under-explored topic. In this paper, we propose the first unsupervised multi-granularity summarization framework, GranuSum. We take events as the basic semantic units of the source documents and propose to rank these events by their salience. We also develop a model to summarize input documents with given events as anchors and hints. By inputting different numbers of events, GranuSum is capable of producing multi-granular summaries in an unsupervised manner. Meanwhile, to evaluate multi-granularity summarization models, we annotate a new benchmark GranuDUC, in which we write multiple summaries of different granularities for each document cluster. Experimental results confirm the substantial superiority of GranuSum on multigranularity summarization over several baseline systems. Furthermore, by experimenting on conventional unsupervised abstractive summarization tasks, we find that GranuSum, by exploiting the event information, can also achieve new state-of-the-art results under this scenario, outperforming strong baselines.