We present the first results from a major Hubble Space Telescope programme designed to investigate the cosmological evolution of quasar host galaxies from z ≃ 2 to the present day. Here we describe J and H-band NICMOS imaging of two quasar samples at redshifts of 0.9 and 1.9 respectively. Each sample contains equal numbers of radioloud and radio-quiet quasars, selected to lie within the same narrow range of optical absolute magnitude (−24 ≥ M V ≥ −25). Filter and target selection were designed to ensure that at each redshift the images sample the same part of the object's restframe spectrum, longwards of 4000Å where starlight from the host galaxy is relatively prominent, but avoiding potential contamination by [Oiii]λ5007 and Hα emission lines.At z ≃ 1 we have been able to establish host-galaxy luminosities and scalelengths with sufficient accuracy to demonstrate that the hosts of both radio-loud and radioquiet quasars lie on the same Kormendy relation described by 3CR radio galaxies at comparable redshift . Taken at face value the gap between the host luminosities of radio-loud and radio-quiet objects appears to have widened from only ≃ 0.4 mag. at z ≃ 0.2 to ≃ 1 mag. at z ≃ 1, a difference that cannot be due to emission-line contamination given the design of our study. However, within current uncertainties, simple passive stellar evolution is sufficient to link these galaxies with the elliptical hosts of low-redshift quasars of comparable nuclear output, implying that the hosts are virtually fully assembled by z ∼ 1.At z ≃ 2 the hosts have proved harder to characterise accurately, and for only two of the nine z ≃ 2 quasars observed has it proved possible to properly constrain the scalelength of the host galaxy. However, the data are of sufficient quality to yield hostgalaxy luminosities accurate to within a factor ≃ 2. At this redshift the luminosity gap between radio-loud and radio-quiet quasars appears to have widened further to ≃ 1.5 mag. Thus while the hosts of radio-loud quasars remain consistent with a formation epoch of z > 3, allowing for passive evolution implies that the hosts of radio-quiet quasars are ≃ 2 − 4 times less massive at z ≃ 2 than at z ≃ 0.2.If the relationship between black-hole and spheroid mass is unchanged out to redshift z ≃ 2, then our results rule out any model of quasar evolution which involves a substantial component of luminosity evolution (e.g. Kauffmann & Haehnelt 2000). Rather, this study indicates that at z ≃ 2 there is a substantial increase in the number density of active black holes, along with a moderate increase in the fueling efficiency of a typical observed quasar. The fact that this latter effect is not displayed by the radio-loud objects in our sample might be explained by a selection effect arising from the fact that powerful radio sources are only produced by the most massive black holes McLure & Dunlop 2000b).