Strong evidence for cooling flows has been found in low-resolution X-ray imaging and spectra of many clusters of galaxies. However, high-resolution X-ray spectra of several clusters from the Reflection Grating Spectrometer on XMM-Newton now show a soft X-ray spectrum inconsistent with a simple cooling flow. The main problem is a lack of the emission lines expected from gas cooling below 1–2 keV. Lines from gas at about 2–3 keV are observed, even in a high-temperature cluster such as A1835, indicating that gas is cooling down to about 2–3 keV, but is not found at lower temperatures. Here we discuss several solutions to the problem: heating, mixing, differential absorption and inhomogeneous metallicity. Continuous or sporadic heating creates further problems, including the targeting of the heat at the cooler gas and also the high total energy required. So far there is no clear observational evidence for widespread heating, or shocks, in cluster cores, except in radio lobes which occupy only part of the volume. Alternatively, if the metals in the intracluster medium are not uniformly spread but are clumped, then little line emission is expected from the gas cooling below 1 keV. The low-metallicity part cools without line emission, whereas the strengths of the soft X-ray lines from the metal-rich gas depend on the mass fraction of that gas and not on the abundance, since soft X-ray line emission dominates the cooling function below 2 keV.