To perform their functions, many DNA-dependent proteins have to quickly locate specific targets against the vast excess of nonspecific DNA. Although this problem was first formulated over 40 years ago, the mechanism of such search remains one of the unsolved fundamental problems in the field of protein-DNA interactions. Several complementary mechanisms have been suggested: sliding, based on one-dimensional random diffusion along the DNA contour; hopping, in which the protein "jumps" between the closely located DNA fragments; macroscopic association-dissociation of the protein-DNA complex; and intersegmental transfer. This review covers the modern state of the problem of target DNA search, theoretical descriptions, and methods of research at the macroscopic (molecule ensembles) and microscopic (individual molecules) levels. Almost all studied DNA-dependent proteins search for specific targets by combined three-dimensional diffusion and one-dimensional diffusion along the DNA contour.