Atomic Force Microscopy (AFM) has a great potential as a tool to characterize mechanical and morphological properties of living cells; these properties have been shown to correlate with cells' fate and patho-physiological state in view of the development of novel early-diagnostic strategies. Although several reports have described experimental and technical approaches for the characterization of cellular elasticity by means of AFM, a robust and commonly accepted methodology is still lacking. Here, we show that micrometric spherical probes (also known as colloidal probes) are well suited for performing a combined topographic and mechanical analysis of living cells, with spatial resolution suitable for a complete and accurate mapping of cell morphological and elastic properties, and superior reliability and accuracy in the mechanical measurements with respect to conventional and widely used sharp AFM tips. We address a number of issues concerning the nanomechanical analysis, including the applicability of contact mechanical models and the impact of a constrained contact geometry on the measured Young's modulus (the finite-thickness effect). We have tested our protocol by imaging living PC12 and MDA-MB-231 cells, in order to demonstrate the importance of the correction of the finite-thickness effect and the change in Young's modulus induced by the action of a cytoskeleton-targeting drug.