Low-light image enhancement algorithms (LIEA) can light up images captured in dark or back-lighting conditions. However, LIEA may introduce various distortions such as structure damage, color shift, and noise into the enhanced images. Despite various LIEAs proposed in the literature, few efforts have been made to study the quality evaluation of low-light enhancement. In this article, we make one of the first attempts to investigate the quality assessment problem of low-light image enhancement. To facilitate the study of objective image quality assessment (IQA), we first build a large-scale low-light image enhancement quality (LIEQ) database. The LIEQ database includes 1,000 light-enhanced images, which are generated from 100 low-light images using 10 LIEAs. Rather than evaluating the quality of light-enhanced images directly, which is more difficult, we propose to use the multi-exposure fused (MEF) image and stack-based high dynamic range (HDR) image as a reference and evaluate the quality of low-light enhancement following a full-reference (FR) quality assessment routine. We observe that distortions introduced in low-light enhancement are significantly different from distortions considered in traditional image IQA databases that are well-studied, and the current state-of-the-art FR IQA models are also not suitable for evaluating their quality. Therefore, we propose a new FR low-light image enhancement quality assessment (LIEQA) index by evaluating the image quality from four aspects: luminance enhancement, color rendition, noise evaluation, and structure preserving, which have captured the most key aspects of low-light enhancement. Experimental results on the LIEQ database show that the proposed LIEQA index outperforms the state-of-the-art FR IQA models. LIEQA can act as an evaluator for various low-light enhancement algorithms and systems. To the best of our knowledge, this article is the first of its kind comprehensive low-light image enhancement quality assessment study.