The nanomechanical properties of living cells, such as their surface elastic response and adhesion, have important roles in cellular processes such as morphogenesis(1), mechano-transduction(2), focal adhesion(3), motility(4,5), metastasis(6) and drug delivery(7-10). Techniques based on quasi-static atomic force microscopy techniques(11-17) can map these properties, but they lack the spatial and temporal resolution that is needed to observe many of the relevant details. Here, we present a dynamic atomic force microscopy(18-28) method to map quantitatively the nanomechanical properties of live cells with a throughput (measured in pixels/minute) that is similar to 10-1,000 times higher than that achieved with quasi- static atomic force microscopy techniques. The local properties of a cell are derived from the 0th, 1st and 2nd harmonic components of the Fourier spectrum of the AFM cantilevers interacting with the cell surface. Local stiffness, stiffness gradient and the viscoelastic dissipation of live Escherichia coli bacteria, rat fibroblasts and human red blood cells were all mapped in buffer solutions. Our method is compatible with commercial atomic force microscopes and could be used to analyse mechanical changes in tumours, cells and biofilm formation with sub-10 nm detail.