Most practically relevant inverse problems are highly ill-posed and a variety of regularization techniques has been developed during the last decades for stably reconstructing useful solutions. Among these techniques are shape based strategies which assume that the medium to recover is composed of various 'zones' of unknown topology and separated by sharp interfaces, each of them potentially with unknown smooth internal parameter profiles. For recovering these zones and the internal profiles variants of a level set technique have been proposed by various authors and for a variety of applications. In this talk, we will present some of our own work in this direction which includes history matching of oil reservoirs from production data, microwave medical imaging for breast tumor detection, crack detection from EIT data, low frequency electromagnetic induction tomography (EMIT) and diffuse optical tomography using a linear transport equation. We will concentrate mostly on 2D simulated cases (except of EMIT which is intrinsically 3D) and will point out cross-disciplinary similarities. For diffuse optical tomography we will also compare level set based reconstructions with a newly developed sparsity promoting regularization strategy using an iterated soft-shrinkage algorithm.