This dissertation addresses the generalization of rigid-body attitude kinematics, dynamics, and control to higher dimensions. A new result is developed that demonstrates the kinematic relationship between the angular velocity in N-dimensions and the derivative of the principal-rotation parameters. A new minimum-parameter description of N-dimensional orientation is directly related to the principal-rotation parameters. The mapping of arbitrary dynamical systems into N-dimensional rotations and the merits of new quasi velocities associated with the rotational motion are studied. A Lagrangian viewpoint is used to investigate the rotational dynamics of N-dimensional rigid bodies through Poincar??e??s equations. The N-dimensional, orthogonal angularvelocity components are considered as quasi velocities, creating the Hamel coefficients. Introducing a new numerical relative tensor provides a new expression for these coefficients. This allows the development of a new vector form of the generalized Euler rotational equations. An N-dimensional rigid body is defined as a system whose configuration can be completely described by an N??N proper orthogonal matrix. This matrix can be related to an N??N skew-symmetric orientation matrix. These Cayley orientation variables and the angular-velocity matrix in N-dimensions provide a new connectionbetween general mechanical-system motion and abstract higher-dimensional rigidbody rotation. The resulting representation is named the Cayley form. Several applications of this form are presented, including relating the combined attitude and orbital motion of a spacecraft to a four-dimensional rotational motion. A second example involves the attitude motion of a satellite containing three momentum wheels, which is also related to the rotation of a four-dimensional body. The control of systems using the Cayley form is also covered. The wealth of work on three-dimensional attitude control and the ability to apply the Cayley form motivates the idea of generalizing some of the three-dimensional results to Ndimensions. Some investigations for extending Lyapunov and optimal control results to N-dimensional rotations are presented, and the application of these results to dynamical systems is discussed. Finally, the nonlinearity of the Cayley form is investigated through computing the nonlinearity index for an elastic spherical pendulum. It is shown that whereas the Cayley form is mildly nonlinear, it is much less nonlinear than traditional spherical coordinates.