This paper presents a new formulation of the kinematics of closed-chain mechanisms and its applications to obtaining the kinematic solutions and analyzing the singularities. Closed-chain mechanisms under consideration may have the redundancy in the number of joints. A closed-chain mechanism can be treated as the parallel connection of two open-chains with respect to a point of interest. The kinematics of a closed-chain mechanism is then obtained by imposing the kinematic constraints of the closed-chain on the kinematics of the two open-chains. First, we formulate the kinematics of a closed-chain mechanism using the kinematic constraint between the controllable active joints and the rest of joints, instead of the kinematic constraint between the two open-chains. The kinematic formulation presented in this paper is valid for closed-chain mechanisms with and without the redundancy. Next, based on the derived kinematics of a closed-chain mechanism, we provide the kinematic solutions which are more physically meaningful and less sensitive to numerical instability, and also suggest an effective way to analyze the singularities. Finally, the computational cost associated with the kinematic formulation is analyzed.