In liquids of high glass-forming ability, in which crystal growth rates are low, the rates can be measured over the full range of supercooling from the liquidus temperature down to the glass transition. For systems of low glass-forming ability, growth rates are readily measured at small supercooling and at very large supercooling around the glass-transition temperature, but it is difficult to acquire data over the full range of intermediate supercooling, especially at the maximum in growth rate. Data at intermediate supercoolings are however of considerable interest for understanding glass formation in such systems as pure metals and chalcogenides for phase-change data storage. We will review the methods emerging for making such measurements, and will note that the fragility of the liquid (including possible crossover from ‘fragile’ to ‘strong’ liquid behaviour on cooling) is an important part of understanding fast crystal growth. We also note that there are deficiencies in existing theories of fast crystal growth.