This paper develops new Newton and conjugate gradient algorithms on the Grassmann and Stiefel manifolds, which are used to solve problems with orthogonality constraints. These manifolds arise in various areas such as symmetric eigenvalue problems, nonlinear eigenvalue problems, electronic structure computations, and signal processing. The authors provide a geometrical framework that unifies these algorithms and offers new insights into existing methods. They derive new formulas for geodesics and parallel translation on the Stiefel and Grassmann manifolds, which are essential for the algorithms. The paper also discusses the canonical metric on these manifolds and provides methods for computing geodesics and parallel translation efficiently. The algorithms are designed to be competitive with existing special algorithms and offer a unified approach to solving problems with orthogonality constraints.This paper develops new Newton and conjugate gradient algorithms on the Grassmann and Stiefel manifolds, which are used to solve problems with orthogonality constraints. These manifolds arise in various areas such as symmetric eigenvalue problems, nonlinear eigenvalue problems, electronic structure computations, and signal processing. The authors provide a geometrical framework that unifies these algorithms and offers new insights into existing methods. They derive new formulas for geodesics and parallel translation on the Stiefel and Grassmann manifolds, which are essential for the algorithms. The paper also discusses the canonical metric on these manifolds and provides methods for computing geodesics and parallel translation efficiently. The algorithms are designed to be competitive with existing special algorithms and offer a unified approach to solving problems with orthogonality constraints.