Hand hygiene is a standard six-step hand-washing action proposed by the World Health Organization (WHO). However, there is no good way to supervise medical staff to do hand hygiene, which brings the potential risk of disease spread. In this work, we propose a new computer vision task called hand hygiene assessment to provide intelligent supervision of hand hygiene for medical staff. Existing action assessment works usually make an overall quality prediction on an entire video. However, the internal structures of hand hygiene action are important in hand hygiene assessment. Therefore, we propose a novel fine-grained learning framework to perform step segmentation and key action scorer in a joint manner for accurate hand hygiene assessment. Existing temporal segmentation methods usually employ multi-stage convolutional network to improve the segmentation robustness, but easily lead to over-segmentation due to the lack of the long-range dependence. To address this issue, we design a multi-stage convolution-transformer network for step segmentation. Based on the observation that each hand-washing step involves several key actions which determine the hand-washing quality, we design a set of key action scorers to evaluate the quality of key actions in each step. In addition, there lacks a unified dataset in hand hygiene assessment. Therefore, under the supervision of medical staff, we contribute a video dataset that contains 300 video sequences with fine-grained annotations. Extensive experiments on the dataset suggest that our method well assesses hand hygiene videos and achieves outstanding performance.