One of the most critical aspects of protein–DNA interactions is the ability of protein molecules to quickly find and recognize specific target sequences on DNA. Experimental measurements indicate that the corresponding association rates to few specific sites among large number of non-specific sites are typically large. For some proteins they might be even larger than maximal allowed three-dimensional diffusion rates. Although significant progress in understanding protein search and recognition of targets on DNA has been achieved, detailed mechanisms of these processes are still strongly debated. Here we present a critical review of current theoretical approaches and some experimental observations in this area. Specifically, the role of lowering dimensionality, non-specific interactions, diffusion along the DNA molecules, protein and target sites concentrations, and electrostatic effects are critically analyzed. Possible future directions and outstanding problems are also presented and discussed.