In recent decades, visual simultaneous localization and mapping (vSLAM) has gained significant interest in both academia and industry. It estimates camera motion and reconstructs the environment concurrently using visual sensors on a moving robot. However, conventional cameras are limited by hardware, including motion blur and low dynamic range, which can negatively impact performance in challenging scenarios like high-speed motion and high dynamic range illumination. Recent studies have demonstrated that event cameras, a new type of bio-inspired visual sensor, offer advantages such as high temporal resolution, dynamic range, low power consumption, and low latency. This paper presents a timely and comprehensive review of event-based vSLAM algorithms that exploit the benefits of asynchronous and irregular event streams for localization and mapping tasks. The review covers the working principle of event cameras and various event representations for preprocessing event data. It also categorizes event-based vSLAM methods into four main categories: feature-based, direct, motion-compensation, and deep learning methods, with detailed discussions and practical guidance for each approach. Furthermore, the paper evaluates the state-of-the-art methods on various benchmarks, highlighting current challenges and future opportunities in this emerging research area. A public repository will be maintained to keep track of the rapid developments in this field at {\url{https://github.com/kun150kun/ESLAM-survey}}.