While in conventional mechanical bearings motion is obtained by sliding or rolling between solid bodies with the resulting mechanical non-linearities, flexible bearings rely on the elastic properties of matter allowing several advantages to be obtained. Combining flexible bearings with adequate actuators and sensors allows thus nanometric positioning resolutions to be obtained. In this paper, the basic principles of compliant mechanisms design methodology are presented. To illustrate the approach, some examples taken from aerospace and astronomical applications are described. Several compliant mechanisms used at the Swiss Light Source synchrotron facility are then presented.