Perceptual Video Quality Assessment (VQA) is one of the most fundamental and challenging problems in the field of Video Engineering. Along with video compression, it has become one of two dominant theoretical and algorithmic technologies in television streaming and social media. Over the last 2 decades, the volume of video traffic over the internet has grown exponentially, powered by rapid advancements in cloud services, faster video compression technologies, and increased access to high-speed, low-latency wireless internet connectivity. This has given rise to issues related to delivering extraordinary volumes of picture and video data to an increasingly sophisticated and demanding global audience. Consequently, developing algorithms to measure the quality of pictures and videos as perceived by humans has become increasingly critical since these algorithms can be used to perceptually optimize trade-offs between quality and bandwidth consumption. VQA models have evolved from algorithms developed for generic 2D videos to specialized algorithms explicitly designed for on-demand video streaming, user-generated content (UGC), virtual and augmented reality (VR and AR), cloud gaming, high dynamic range (HDR), and high frame rate (HFR) scenarios. Along the way, we also describe the advancement in algorithm design, beginning with traditional hand-crafted feature-based methods and finishing with current deep-learning models powering accurate VQA algorithms. We also discuss the evolution of Subjective Video Quality databases containing videos and human-annotated quality scores, which are the necessary tools to create, test, compare, and benchmark VQA algorithms. To finish, we discuss emerging trends in VQA algorithm design and general perspectives on the evolution of Video Quality Assessment in the foreseeable future.