We present the results of a deep K-band imaging study which reveals the host galaxies around a sample of luminous radio-quiet quasars. The K-band images, obtained at UKIRT, are of sufficient quality to allow accurate modelling of the underlying host galaxy. Initially, the basic structure of the hosts is revealed using a modified clean deconvolution routine optimized for this analysis. Two of the 14 quasars are shown to have host galaxies with violently disturbed morphologies which cannot be modelled by smooth elliptical profiles. For the remainder of our sample, 2D models of the host and nuclear component are fitted to the images using the χ2 statistic to determine goodness of fit. Host galaxies are detected around all of the quasars. The reliability of the modelling is extensively tested, and we find the host luminosity to be well constrained for nine quasars. The derived average K-band absolute K-corrected host galaxy magnitude for these luminous radio-quiet quasars is =25.15+/-0.04, slightly more luminous than an L* galaxy. The spread of derived host galaxy luminosities is small, although the spread of nuclear-to-host ratios is not. These host luminosities are shown to be comparable to those derived from samples of quasars of lower total luminosity, and we conclude that there is no correlation between host and nuclear luminosity for these quasars. Nuclear-to-host ratios break the lower limit previously suggested from studies of lower nuclear luminosity quasars and Seyfert galaxies. Morphologies are less certain but, on the scales probed by these images, some hosts appear to be dominated by spheroids while others appear to have disc-dominated profiles.