Safety constraints are ubiquitous in many robotic applications. For instance, aerial robots such as quadrotors or hexcoptors need to realize fast collision-free flight, and bipedal robots have to choose their discrete footholds properly to gain the desired friction and pressure contact forces. In this thesis, we address the safety critical control problem for fully-actuated and under-actuated mechanical systems. Since many mechanical systems evolve on nonlinear manifolds, we extend the concept of Control Barrier Function to a new concept called geometric Control Barrier Function which is specifically designed to handle safety constraints on manifolds. This type of Control Barrier Function stems from geometric control techniques and has a coordinate free and compact representation. In a similar fashion, we also extend the concept of Control Lyapunov Function to the concept of geometric Control Lyapunov Function to realize tracking on the manifolds. Based on these new geometric versions of CLF and CBF, we propose a general control design method for fully-actuated systems with both state and input constraints. In this CBF-CLF-QP control design, the control input is computed based on a state-dependent Quadratic Programming (QP) where the safety constraints are strictly enforced using geometric CBF but the tracking constraint is imposed through a type of relaxation. Through this type of relaxation, the controller could still keep the system state safe even in the cases when the reference is unsafe during some time period. For a single quadrotor, we propose the concept of augmented Control Barrier Function specifically to let it avoid external obstacles. Using this augmented CBF, we could still utilize the idea of CBF-CLF-QP controller in a sequential QP control design framework to let this quadrotor remain safe during the flight. In meantime, we also apply the geometric control techniques to the aerial transportation problem where a payload is carried by multiple quadrotors through cable suspension. This type of transportation method allows multiple quadrotors to share the payload weight, but introduces internal safety constraints at the same time. By employing both linear and nonlinear techniques, we are able to carry the payload pose to follow a pre-defined reference trajectory.