The relationship between biological cells' mechanical behavior and their underlying molecular structures and cellular dynamics opens new perspectives for polymer physics. Intracellular organization and cellular stability rely on an active polymer scaffold, the cytoskeleton, which is unparalleled in the synthetic world. Here, we highlight micro- and nanotechnology-based methods that have been used to study the active and passive biomechanical properties of living cells. Correlating viscoelastic properties with the metastatic potential of the cells could produce new insights into the processes of cell migration and metastasis.