This article proposes a versatile quadruped climbing robot (QCR) based on multidegree-of-freedom legs utilizing adhesive vacuum suckers. For the first time, the climbing robot is simultaneously endowed with climbing, omnidirectional locomotion, large obstacle overcoming, and manipulation capabilities. A prototype QCR is developed by integrating the functional components along with vision and torque sensors. The kinematic and quasi-static climbing models have been derived under different scenarios with climbing safety checks by analyzing the adhesive and surface reaction forces. For potential application on the glass façade of high-rise buildings that are slippery and often full of prominent structures, a forward crawling gait and an omnidirectional climbing gait are planned to overcome the obstacles. A comprehensive torque controller is proposed for locomotion. The concept of the leg-transformed manipulator is introduced to accomplish the manipulation tasks under visual guidance. The robot's performance has been verified by performing several experiments in adhesion, crawling, visual-guided manipulation, and climbing. Results reveal that the developed QCR can stably climb, overcome obstacles, and perform manipulation on surfaces with different inclination angles and obstacles.