Fog removal from an image is an active research topic in computer vision. However, current literature is weak in the following two areas which in many ways are hindering progress for developing defogging algorithms. First, there is no true real-world and naturally occurring foggy image datasets suitable for developing defogging models. Second, there is no suitable mathematically simple and easy to use image quality assessment (IQA) methods for evaluating the visual quality of defogged images. We address these two aspects in this paper. We first introduce a new foggy image dataset called multiple real-world foggy image dataset (MRFID). MRFID contains foggy and clear images of 200 outdoor scenes. For each scene, one clear image and 4 foggy images of different densities defined as slightly foggy, moderately foggy, highly foggy, and extremely foggy, are manually selected from images taken from these scenes over the course of one calendar year. We then process the foggy images of MRFID using 16 defogging methods to obtain 12,800 defogged images (DFIs) and perform a comprehensive subjective evaluation of the visual quality of the DFIs. Through collecting the mean opinion score (MOS) of 120 subjects and evaluating a variety of fog-relevant image features, we have developed a new Fog-relevant Feature based SIMilarity index (FRFSIM) for assessing the visual quality of DFIs. We present extensive experimental results to show that our new visual quality assessment measure, the FRFSIM, is more consistent with the MOS than other IQA methods and is therefore more suitable for evaluating defogged images than other state-of-the-art IQA methods. Our dataset and relevant code are available at http://www.vistalab.ac.cn/MRFID-for-defogging/.