In these lectures we review some recent results concerning inverse problems for thin elastic nanostructures. Nanostructures are assumed to be either one-dimensional (nanobeams) or two-dimensional (nanoplates), and are described within a simplified version of the strain gradient linear elasticity theory for isotropic materials. A first group of results concerns the use of nanobeams as mass-resonant sensors to identify an unknown added mass density by the measurement of a finite number of lower resonant frequencies. In the second group of results, we determine constructive upper and lower estimates of the area of an unknown elastic inclusion possibly present in a nanoplate given in terms of the work exerted by force and couple fields applied at the boundary of the nanoplate.