Video quality assessment (VQA) plays a vital role in the field of video processing, including areas of video acquisition, video filtering in retrieval, video compression, video restoration, and video enhancement. Since VQA has gained much attention in recent years, this paper gives an up-to-date review of VQA research and highlights current challenges in this filed. The subjective study and common VQA databases are first reviewed. Then, a survey on the objective VQA methods, including full-reference, reduced-reference, and no-reference VQA, is reported. Last but most importantly, the key limitations of current research and several challenges in the field of VQA are discussed, which include the impact of video content, memory effects, computational efficiency, personalized video quality prediction, and quality assessment of newly emerged videos.