Studies of the deformation characteristics of single biological cells can offer insights into the connections among mechanical state, biochemical response and the onset and progression of diseases. Deformation imposed by optical tweezers provides a useful means for the study of single cell mechanics under a variety of well-controlled stress-states. In this paper, we first critically review recent advances in the study of single cell mechanics employing the optical tweezers method, and assess its significance and limitations in comparison to other experimental tools. We then present new experimental and computational results on shape evolution, force-extension curves, elastic properties and viscoelastic response of human red blood cells subjected to large elastic deformation using optical tweezers. Potential applications of the methods examined here to study diseased cells are also briefly addressed.