We give a survey of research on the problem of single-weighted and double-weighted
estimates of strong and weak types for the Hardy — Littlewood maximal function, Riesz
potentials, singular integral operators, and harmonic functions. Necessary and
sufficient conditions on the weight are given under which weighted estimates are
valid (Muckenhoupt's Ap-condition, Sawyer's condition, etc.). Special attention is
given to papers which appeared after 1980 and the latest results, published as reports
and preprints.