Banding, also known as staircase-like contours, frequently occurs in flat areas of images/videos processed by the compression or quantization algorithms. As undesirable artifacts, banding destroys the original image structure, thus degrading users' quality of experience (QoE). In this paper, we systematically investigate the banding image quality assessment (IQA) problem, aiming to detect the image banding artifacts and evaluate their perceptual visual quality. Considering that the existing image banding databases only contain limited content sources and banding generation methods, and lack perceptual quality labels (i.e. mean opinion scores), we first build the largest banding IQA database so far, named Banding Artifact Noticeable Database (BAND-2k), which consists of 2,000 banding images generated by 15 compression and quantization schemes. A total of 23 workers participated in the subjective IQA experiment, yielding over 214,000 patch-level banding class labels and 44,371 reliable image-level quality ratings. Subsequently, we develop an effective no-reference (NR) banding evaluator for banding detection and quality assessment by leveraging frequency characteristics of banding artifacts. A dual convolutional neural network is employed to concurrently learn the feature representation from the high-frequency and low-frequency maps, thereby enhancing the ability to discern banding artifacts. The quality score of a banding image is generated by pooling the banding detection maps masked by the spatial frequency filters. Experiments demonstrate that our banding evaluator achieves a remarkably high accuracy in banding detection and also exhibits high SRCC and PLCC results with the perceptual quality labels. These findings unveil the strong correlations between the intensity of banding artifacts and the perceptual visual quality, thus validating the necessity of banding quality assessment.