Obstacle avoidance for Unmanned Aerial Vehicles (UAVs) in cluttered environments is significantly challenging. Existing obstacle avoidance for UAVs either focuses on fully static environments or static environments with only a few dynamic objects. In this paper, we take the initiative to consider the obstacle avoidance of UAVs in dynamic cluttered environments in which dynamic objects are the dominant objects. This type of environment poses significant challenges to both perception and planning. Multiple dynamic objects possess various motions, making it extremely difficult to estimate and predict their motions using one motion model. The planning must be highly efficient to avoid cluttered dynamic objects. This paper proposes Fast and Adaptive Perception and Planning (FAPP) for UAVs flying in complex dynamic cluttered environments. A novel and efficient point cloud segmentation strategy is proposed to distinguish static and dynamic objects. To address multiple dynamic objects with different motions, an adaptive estimation method with covariance adaptation is proposed to quickly and accurately predict their motions. Our proposed trajectory optimization algorithm is highly efficient, enabling it to avoid fast objects. Furthermore, an adaptive re-planning method is proposed to address the case when the trajectory optimization cannot find a feasible solution, which is common for dynamic cluttered environments. Extensive validations in both simulation and real-world experiments demonstrate the effectiveness of our proposed system for highly dynamic and cluttered environments.