This thesis presents a geometric approach to studying kinematics, dynamics and controls of open-chain multi-body systems with non-zero momentum and multi-degree-of-freedom joints subject to holonomic and nonholonomic constraints. Some examples of such systems appear in space robotics, where mobile and free-base manipulators are developed. The proposed approach introduces a unified framework for considering holonomic and nonholonomic, multi-degree-of-freedom joints through: (i) generalization of the product of exponentials formula for kinematics, and (ii) aggregation of the dynamical reduction theories, using differential geometry. Further, this framework paves the ground for the input-output linearization and controller design for concurrent trajectory tracking of base-manipulator(s). In terms of kinematics, displacement subgroups are introduced, whose relative configuration manifolds are Lie groups and they are parametrized using the exponential map. Consequently, the product of exponentials formula for forward and differential kinematics is generalized to include multi-degree-of-freedom joints and nonholonomic constraints in open-chain multi-body systems. As for dynamics, it is observed that the action of the relative configuration manifold corresponding to the first joint of an open-chain multi-body system leaves Hamilton's equation invariant. Using the symplectic reduction theorem, the dynamical equations of such systems with constant momentum (not necessarily zero) are formulated in the reduced phase space, which present the system dynamics based on the internal parameters of the system. In the nonholonomic case, a three-step reduction process is presented for nonholonomic Hamiltonian mechanical systems. The Chaplygin reduction theorem eliminates the nonholonomic constraints in the first step, and an almost symplectic reduction procedure in the unconstrained phase space further reduces the dynamical equations. Consequently, the proposed approach is used to reduce the dynamical equations of nonholonomic open-chain multi-body systems. Regarding the controls, it is shown that a generic free-base, holonomic or nonholonomic open-chain multi-body system is input-output linearizable in the reduced phase space. As a result, a feed-forward servo control law is proposed to concurrently control the base and the extremities of such systems. It is shown that the closed-loop system is exponentially stable, using a proper Lyapunov function. In each chapter of the thesis, the developed concepts are illustrated through various case studies.