Summary: We describe the main results on the uniqueness and stability of penetrable or impenetrable obstacles from various boundary data. We present basic ideas of proofs in the case of single and many boundary measurements. We discuss some methods of reconstruction. We consider the inverse problem of gravimetry, inverse conductivity and scattering problems, and similar inverse obstacle problems for hyperbolic and parabolic equations. We expose mostly known results and some new ideas and approaches. In conclusion, we list several open research problems of significance for theory and applications.