Indoor service robots have been widely introduced in the fields of cleaning, delivery, education, guidance, and healthcare, etc. in indoor environments. The mobility of an indoor service robot is essential to expanding its applications. However, the mobility of existing indoor service robots is highly limited by surrounding indoor environments. For example, a stair is one of the major obstacles that restrict the reachable areas of indoor service robots. Even though many studies have been performed to develop reliable and fast stair-climbing robots based on legged, tracked, wheel-legged and wheel-linkage mechanisms, a market-dominant stair-climbing robot remains unsolved. This review investigates the efforts of engineers devoted to stair-climbing robots. To this end, the locomotion mechanisms of stair climbing robots are classified and their sensing method are summarized. In this review, we also propose useful criteria for evaluating the stair-climbing ability of an indoor service robot. By virtue of these criteria, the stair-climbing performance of existing robots are qualitatively compared. We hope this review helps develop the reliable and fast stair-climbing robots and the reasonable criteria for their performances.