The field of perceptual quality assessment has gone through a wide range of developments and it is still growing. In particular, the area of no-reference (NR) image and video quality assessment has progressed rapidly during the last decade. In this article, we present a classification and review of latest published research work in the area of NR image and video quality assessment. The NR methods of visual quality assessment considered for review are structured into categories and subcategories based on the types of methodologies used for the underlying processing employed for quality estimation. Overall, the classification has been done into three categories, namely, pixel-based methods, bitstream-based methods, and hybrid methods of the aforementioned two categories. We believe that the review presented in this article will be helpful for practitioners as well as for researchers to keep abreast of the recent developments in the area of NR image and video quality assessment. This article can be used for various purposes such as gaining a structured overview of the field and to carry out performance comparisons for the state-of-the-art methods.