This is an advanced textbook based on lectures delivered at the Moscow Physico-Technical Institute. Brevity, logical organization of the material, and a sometimes lighthearted approach are distinctive features of this compact book. The author turns the reader into an active participant by asking questions, hinting, giving direct recommendations, comparing different methods, and discussing "pessimistic" and "optimistic" approaches to numerical analysis. Since matrix analysis underlies numerical methods and the author is an expert in this field, emphasis in the book is on methods and algorithms of matrix analysis. Also considered are function approximations, methods of solving nonlinear equations and minimization methods. Alongside classical methods, new results and approaches developed over the last few years are discussed - namely those on spectral distribution theory and design and proof of modern preconditioning strategies for large-scale linear algebra problems. Advanced students and graduate students majoring in computer science, physics and mathematics will find this book helpful. It can be equally useful for advanced readers and researchers, providing them with new findings and new accessible views of the basic mathematical framework.