Essentially all genetic events require proteins to move from one location in a DNA polymer to another location in the same chain. A protein will seldom bind to a specific site in the DNA by colliding directly with that site. Instead, the protein will almost always collide first with a random site anywhere in the DNA and then migrate to the specific site by a facilitated-diffusion process that is constrained to the zone of that DNA molecule. Thereafter, many proteins bound to their target sites translocate in a specified direction along the DNA by a energy-dependent vectorial mechanism. This review will discuss some of the strategies that have been developed to analyse the motion of proteins on DNA, with respect to both the random diffusion processes involved in target-site location by DNA-binding proteins and the vectorial processes involved in unidirectional translocation along DNA.