In recent years, legged and wheeled-legged robots have gained prominence for tasks in environments predominantly created for humans across various domains. One significant challenge faced by many of these robots is their limited capability to navigate stairs, which hampers their functionality in multi-story environments. This study proposes a method aimed at addressing this limitation, employing reinforcement learning to develop a versatile controller applicable to a wide range of robots. In contrast to the conventional velocity-based controllers, our approach builds upon a position-based formulation of the RL task, which we show to be vital for stair climbing. Furthermore, the methodology leverages an asymmetric actor-critic structure, enabling the utilization of privileged information from simulated environments during training while eliminating the reliance on exteroceptive sensors during real-world deployment. Another key feature of the proposed approach is the incorporation of a boolean observation within the controller, enabling the activation or deactivation of a stair-climbing mode. We present our results on different quadrupeds and bipedal robots in simulation and showcase how our method allows the balancing robot Ascento to climb 15cm stairs in the real world, a task that was previously impossible for this robot.