Living cells are complex mechanical systems and their mechanical properties are involved in various processes in health and disease. Scanning probe microscopy techniques such as the atomic force microscope are powerful tools for investigating cell mechanics on the single-cell and subcellular level. In this talk I will present current advances in the scanning-probe-based investigation of active and passive mechanical properties of living cells. Their exhibit a unique viscoelastic mechanical behavior, which is caused by the disordered active structure of the cytoskeleton and closely related to the acto-myosin contractility. Further experimental and theoretical work is required for a comprehensive understanding of cell mechanics, but scanning probe microscopes and their combination with other techniques are powerful tools for this purpose.