Solid tumours are often first diagnosed by palpation, suggesting that the tumour is more rigid than its surrounding environment. Paradoxically, individual cancer cells appear to be softer than their healthy counterparts. In this review, we first list the physiological reasons indicating that cancer cells may be more deformable than normal cells. Next, we describe the biophysical tools that have been developed in recent years to characterise and model cancer cell mechanics. By reviewing the experimental studies that compared the mechanics of individual normal and cancer cells, we argue that cancer cells can indeed be considered as softer than normal cells. We then focus on the intracellular elements that could be responsible for the softening of cancer cells. Finally, we ask whether the mechanical differences between normal and cancer cells can be used as diagnostic or prognostic markers of cancer progression.