Obstacle identification problems are an important class of inverse problems, where the goal is to find the location, size and shape of a local "object" (which can be a cavity, a region made of material different from the background material, a geometric irregularity in the boundary, etc.) in an otherwise given medium, based on some measurements of the relevant field. A sub-class of problems in this category is that based on time-dependent waves. Here, a given time-dependent wave source is introduced in the medium, and the response to this source is measured at certain points in space and time. Based on these measurements, some computational method is used to identify the obstacle. Such problems appear (and are important), for example, in bio-medical engineering, Non-Destructing Testing (NDT) of structures, damage evaluation, underwater acoustics and solid earth geophysics (SEG). The most common, which is also the simplest, method used for obstacle identification is Arrival Time Imaging (ATI), also called Kirchhoff Migration. It is claimed that about 80% of SEG identification problems are solved using ATI. At the other end of the spectrum there is Full Waveform Inversion (FWI), usually with the aid of an efficient adjoint-type scheme. Computational methods based on Time-Reversal (TR) are also effective for such problems. In this talk we present various recently developed methods to solve different time-dependent wave-based obstacle identification problems in acoustics, SEG and structural damage evaluation. This work is partly based on the recent papers [1-3], and includes insight into single-field identification (reasons for success and failure), an ATI method for elastodynamics, and a precise shape identification method for elastic inclusions. REFERENCES [1] D. Rabinovich, D. Givoli and E. Turkel, “Single-Field Identification of Inclusions and Cavities in an Elastic Medium,'' Int. J. Numerical Methods in Engineering, Vol. 125, pp. e7364-1-29, 2024. [2] D. Rabinovich and D. Givoli, “A Kirchhoff Migration Scheme for Elastic Obstacle Identification,'' Inverse Problems, Vol. 40, pp. 105006-1-24, 2024. [3] A. Sayag and D. Givoli, “Shape and Property Identification of an Elastic Inclusion via a FWI-Adjoint Method,'' J. Theoretical & Computational Acoustics, in press, 2025.