Image segmentation is one of the fundamental problems in computer vision and image processing. In the recent years mathematical models based on partial differential equations and variational methods have led to superior results in many applications, e.g., medical imaging. A majority of works on image segmentation implicitly assume the given image to be biased by additive Gaussian noise, for instance the popular Mumford-Shah model. Since this assumption is not suitable for a variety of problems, we propose a region-based variational segmentation framework to segment also images with non-Gaussian noise models. Motivated by applications in biomedical imaging, we discuss the cases of Poisson and multiplicative speckle noise intensively. Analytical results such as the existence of a solution are verified and we investigate the use of different regularization functionals to provide a-priori information regarding the expected solution. The performance of the proposed framework is illustrated by experimental results on synthetic and real data.