Recognition and binding of specific sites on DNA by proteins is central for many cellular functions such as transcription, replication, and recombination. In the process of recognition, a protein rapidly searches for its specific site on a long DNA molecule and then strongly binds this site. Here we aim to find a mechanism that can provide both a fast search (1-10 s) and high stability of the specific protein-DNA complex (Kd=10(-15)-10(-8) M). Earlier studies have suggested that rapid search involves sliding of the protein along the DNA. Here we consider sliding as a one-dimensional diffusion in a sequence-dependent rough energy landscape. We demonstrate that, despite the landscape's roughness, rapid search can be achieved if one-dimensional sliding is accompanied by three-dimensional diffusion. We estimate the range of the specific and nonspecific DNA-binding energy required for rapid search and suggest experiments that can test our mechanism. We show that optimal search requires a protein to spend half of its time sliding along the DNA and the other half diffusing in three dimensions. We also establish that, paradoxically, realistic energy functions cannot provide both rapid search and strong binding of a rigid protein. To reconcile these two fundamental requirements we propose a search-and-fold mechanism that involves the coupling of protein binding and partial protein folding. The proposed mechanism has several important biological implications for search in the presence of other proteins and nucleosomes, simultaneous search by several proteins, etc. The proposed mechanism also provides a new framework for interpretation of experimental and structural data on protein-DNA interactions.