Uniqueness and stability estimates are two central research questions in inverse scattering problems. A famous question raised by M. Schiffer is whether an obstacle can be uniquely determined by a single far-field measurement. This type of geometric inverse problem is highly challenging and possesses a rich history in the field. Here, a single far-field measurement refers to data obtained at a fixed wavenumber and a fixed incident wave, with observations taken over the entire unit sphere. In this talk, I will discuss the uniqueness and stability aspects of identifying obstacles using a finite number of far-field measurements. Uniqueness is primarily contingent upon the unique geometric structures exhibited by Laplacian eigenfunctions near corners. For the stability estimate, we present a recent result establishing a single stability estimate for determining a general, generalized impedance polytope obstacle using only a single far-field measurement.