Motion planning in dynamic environments requires robots to continuously adapt their paths in response to environmental changes for safe and uninterrupted navigation. While many surveys have reviewed planning in static settings, systematic reviews focused on dynamic environments remain limited. This paper presents a comprehensive survey of 138 works, primarily published between 2015 and 2025, spanning both classical and learning-based approaches. The motion planning methods are grouped into five categories based on the concepts of sampling, graph search, model predictive control, learning, and additional classical local planning approaches, including velocity obstacles, potential fields and dynamic windows. The learning techniques include supervised learning and reinforcement learning. We also discuss the role of dynamic perception in motion planning, covering techniques for detecting and modeling moving obstacles using cameras, LiDAR, and event-based sensors. The survey analyzes the principles, strengths, and limitations of each method, with particular attention to challenges unique to dynamic environments, such as prediction uncertainty, human-robot interaction, and the freezing robot problem. The survey provides researchers with a structured understanding of motion planning methods in dynamic environments.