We present a detailed analysis of the X-ray properties of the cooling flows in a sample of nearby, X-ray-bright clusters of galaxies using high-quality ASCA spectra and ROSAT X-ray images. We demonstrate the need for multiphase models to consistently explain the spectral and imaging X-ray data for the clusters. The mass deposition rates of the cooling flows, independently determined from the ASCA spectra and ROSAT images, exhibit reasonable agreement. We confirm the presence of intrinsic X-ray absorption in the clusters using a variety of spectral models. We also report detections of 100-μm infrared emission, spatially coincident with the cooling flows, in several of the systems studied. The observed infrared fluxes and flux limits are in good agreement with the predicted values owing to reprocessed X-ray emission from the cooling flows. We present precise measurements of the abundances of iron, magnesium, silicon and sulphur in the central regions of the Virgo and Centaurus clusters. Our results firmly favour models in which a high mass fraction (70–80 per cent) of the iron in the X-ray gas in these regions originates from Type Ia supernovae. Finally, we present a series of methods which may be used to estimate the ages of cooling flows from X-ray data. The results for the present sample of clusters indicate ages of between 2.5 and 7 Gyr. If the ages of cooling flows are primarily set by subcluster merger events, then our results suggest that in the largest clusters, mergers with subclusters with masses of ∼30 per cent of the final cluster mass are likely to disrupt cooling flows.