We propose a novel perception-aware planning and control scheme for quadruepdal stair climbing. The quadruped is able to autonomously climb various real-world staircases with dynamical gaits. All computations are done online and using only on-board processor. No external inputs from human-operated remote controller and no additional parameter tuning are required for different staircases. Exploiting prior structural knowledge about staircases, an extremely simple yet effective approach is developed for extracting staircase information from perceived camera data. A novel planning strategy for motion reference during stair climbing is proposed. Combined with an improved model-predictive controller, the quadruped is capable of climbing different staircases fully autonomously, requiring no external commands and no further parameter tuning.