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For various applications, sensors are deployed to monitor belt regions to guarantee that every movement crossing a barrier of sensors will be detected in real-time with high accuracy and minimize the need for human support. The barrier coverage problem is introduced to model these requirements, and has been examined thoroughly in the past decades. In this survey, we state the problem definitions and systematically consider sensing models, design issues and challenges in barrier coverage problem. We also review representative algorithms in this survey. Furthermore, we provide discussions on some extensions and variants of barrier coverage problems.