Minimal and orthogonal residual methods and their generalizations for solving linear operator equations This thesis is concerned with the solution of linear operator equations by projection methods known as minimal residual (MR) and orthogonal residual (OR) methods. We begin with a rather abstract framework of approximation by orthogonal and oblique projection in Hilbert space. When these approximation schemes are applied to sequences of nested spaces with a simple requirement relating trial and test spaces in case of the OR method one can derive at this rather general level the basic relations which have been proved for many specific Krylov subspace methods for solving linear systems of equations in the literature. The crucial quantities with which we describe the behavior of these methods are angles between subspaces. By replacing the given inner product with one that is basis dependent one can also incorporate methods based on non orthogonal bases such as those based on the non Hermitian Lanczos process for solving linear systems.