LiDAR-based place recognition (LPR) is one of the most crucial components of autonomous vehicles to identify previously visited places in GPS-denied environments. Most existing LPR methods use mundane representations of the input point cloud without considering different views, which may not fully exploit the information from LiDAR sensors. In this paper, we propose a cross-view transformer-based network, dubbed CVTNet, to fuse the range image views (RIVs) and bird's eye views (BEVs) generated from the LiDAR data. It extracts correlations within the views themselves using intra-transformers and between the two different views using inter-transformers. Based on that, our proposed CVTNet generates a yaw-angle-invariant global descriptor for each laser scan end-to-end online and retrieves previously seen places by descriptor matching between the current query scan and the pre-built database. We evaluate our approach on three datasets collected with different sensor setups and environmental conditions. The experimental results show that our method outperforms the state-of-the-art LPR methods with strong robustness to viewpoint changes and long-time spans. Furthermore, our approach has a good real-time performance that can run faster than the typical LiDAR frame rate. The implementation of our method is released as open source at: https://github.com/BIT-MJY/CVTNet.