In the talk we present several practically relevant inverse problems arising in medical and geophysical applications where the unknowns of interest can be described by shapes or regions with clear interfaces either between them or against the background. These situations often are difficult to handle with standard pixel- or voxel-based reconstruction algorithms where interfaces typically are severely blurred due to the implicit or explicit regularization of those schemes. Shape-based methods incorporate the presence of interfaces explicitly into the model, which facilitates the reconstruction in situations where prior information suggests that such interfaces are present. We will focus in particular on the performance of the level set technique in these applications, which comes with the numerical advantage that changes in topology are automatically taken into account during the reconstruction loop.