This book deals first with Haar bases, Faber bases and Faber frames for weighted function spaces on the real line and the plane. It extends results in the author's book, "Bases in Function Spaces, Sampling, Discrepancy, Numerical Integration" (EMS, 2010), from unweighted spaces (preferably in cubes) to weighted spaces. The obtained assertions are used to study sampling and numerical integration in weighted spaces on the real line and weighted spaces with dominating mixed smoothness in the plane. A short chapter deals with the discrepancy for spaces on intervals. A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.