The monograph outlines quaternion models and methods of dynamics, inertial navigation and motion control using hypercomplex variables - Hamilton quaternions - to describe motion. A quaternion method for regularization of differential equations of the spatial problem of two bodies and the perturbed central motion of a material point is described, quaternion regular models of celestial mechanics and astrodynamics are given. Quaternion models of the relative motion of dynamically symmetric mechanical systems, the theory of gyroscopes, inertial navigation are considered. With their use, a number of problems of the dynamics of a solid body are solved, the movement of gyro-pendulum systems is studied. Quaternion models and methods for the synthesis of nonlinear laws of control of angular motion of a solid body are described, the problems of the optimal meeting of a controlled spacecraft with an uncontrolled spacecraft are investigated.