Cross-domain recommendation (CDR) plays a critical role in alleviating the sparsity and cold-start problem and substantially boosting the performance of recommender systems. Existing CDR methods prefer to either learn a common preference bridge shared by all users or a personalized preference bridge tailored for each user to transfer user preference from the source domain to the target domain. Although these methods significantly improve the recommendation performance, there are still some limitations. First, these methods usually assume a user only has a unique interest, while ignoring the fact that a user may interact with items with different interest preferences. Second, they learn transformed preference representation mainly relies on the source domain signals, while neglecting the rich information available in the target domain. To handle these issues, in this paper, we propose a novel method named Multi-interest Meta Network with Multi-granularity Target-guided Attention (MIMNet) for cross-domain recommendation. To be specific, we employ the capsule network to learn user multiple interests in the source domain, which will be fed into a meta network to generate multiple interest-level preference bridges. Then, we transfer user representations from the source domain to the target domain based on these multi-interest bridges. In addition, we introduce both fine-grained and coarse-grained target signals to aggregate user transformed interest-level representations by incorporating a novel multi-granularity target-guided attention network. We conduct extensive experimental results on three real-world CDR tasks, and the results show that our proposed approach MIMNet consistently outperforms all baseline methods. The source code of MIMNet is released at https://github.com/marqu22/MIMNet.