Movement is intrinsic to life. Biologists have established that most forms of directed nanoscopic, microscopic and, ultimately, macroscopic movements are powered by molecular motors from the dynein, myosin and kinesin superfamilies. These motor proteins literally walk, step by step, along polymeric filaments, carrying out essential tasks such as organelle transport. In the last few years biological molecular walkers have inspired the development of artificial systems that mimic aspects of their dynamics. Several DNA-based molecular walkers have been synthesised and shown to walk directionally along a track upon sequential addition of appropriate chemical fuels. In other studies, autonomous operation--i.e. DNA-walker migration that continues as long as a complex DNA fuel is present--has been demonstrated and sophisticated tasks performed, such as moving gold nanoparticles from place-to-place and assistance in sequential chemical synthesis. Small-molecule systems, an order of magnitude smaller in each dimension and 1000× smaller in molecular weight than biological motor proteins or the walker systems constructed from DNA, have also been designed and operated such that molecular fragments can be progressively transported directionally along short molecular tracks. The small-molecule systems can be powered by light or chemical fuels. In this critical review the biological motor proteins from the kinesin, myosin and dynein families are analysed as systems from which the designers of synthetic systems can learn, ratchet concepts for transporting Brownian substrates are discussed as the mechanisms by which molecular motors need to operate, and the progress made with synthetic DNA and small-molecule walker systems reviewed (142 references).