Motion planning for a robotic system addresses the problem of finding trajectory and actuator forces that are consistent with a given set of constraints and perform a desired task. In general, the problem is under determined and admits a large number of solutions. The main claim of this dissertation is that a natural way to resolve the indeterminacy is to define performance of a motion and find a solution with the best performance. The motion planning problem is thus formulated as a variational problem. The proposed approach is continuous in the sense that the motion planning problem is not discretized. A distinction is made between dynamic and kinematic motion planning. Dynamic motion planning provides the actuator forces as part of the motion plan and requires finding a motion that is consistent with the dynamic equations of the system, satisfies a given set of equality and inequality constraints, and minimizes a chosen cost functional. In kinematic motion planning, dynamic equations of the system are not taken into account and it is therefore simpler. For dynamic motion planning, a novel numerical method for solving variational problems is developed. The continuous problem is discretized by finite element methods and techniques from nonlinear programming are used to solve the resulting finite dimensional optimization problem. The method is employed to find smooth trajectories and actuator forces for two planar cooperating manipulators holding an object. The computed trajectories are used to model human motions in a two-arm manipulation task. The method is then extended for systems that change the dynamic equations as they move. An example of a simple grasping task illustrates that for such systems variational approach unifies motion planning and task planning. Kinematic motion planning is formulated as a variational problem on the group of spatial rigid body displacements, SE(3). A Riemannian metric and an affine connection are introduced to define cost functionals that measure smoothness of trajectories. Metrics and connections that are important for kinematic analysis are identified. It is then shown how the group structure of SE (3) can be used to find smooth trajectories that satisfy boundary conditions on positions, orientations, velocities or their derivatives, and have certain invariance properties with respect to the choice of the inertial and body fixed frames.