Contents: MOTIVATION -- Non-linear elliptic equations in model problems; Linear algebraic systems; Linear elliptic problems; Non-linear algebraic systems and preconditioning. THEORETICAL BACKGROUND -- Non-linear equations in Hilbert space; Solvability of non-linear elliptic problems. ITERATIVE SOLUTION OF NON-LINEAR ELLIPTIC BOUNDARY VALUE PROBLEMS -- Iterative methods in Sobolev space; Preconditioning strategies for discretise non-linear elliptic problems based on preconditioning operators; Algorithmic realisation of iterative methods based on pre-conditioning operators; Some numerical algorithms for non-linear elliptic problems in physics; Appendix; Index.