The number of images captured every day increases as time goes on. Many of these images are taken outdoors, where weather effects can influence the quality of the captured image. The degradation caused by atmospheric effects can be problematic, both in terms of visibility for a human observer and reducing the efficacy of computer vision algorithms. Because of this, the task of restoring these images is essential. This thesis focuses on restoring images degraded by fog or rain. The current state-of-the-art methods in the field are all based on supervised learning. This is a problem because supervised learning is done on synthetic datasets, resulting in poor generalization to real data. We propose a variational model for single-image dehazing based on a smooth approximation of the dark channel prior. We also formulate an unsupervised deep learning model for dehazing based on the same functional. We also propose a variational model for single-image rain removal. The proposed models do not require synthetic data, so they have better properties on real hazy and rainy images than the state-of-the-art models. We compared the proposed methods to the state-of-the-art on real images degraded by weather, and our methods proved to be better in restoring the images. The comparison was made subjectively by visual inspection and objectively by comparing the improvement in object detection performance of restored images. Contributions: Optimization based method for single color image haze removal based on variational formulation of the assumption on minimal value of color component and a biased object depth estimator Optimization based method for single image rain removal with loss function based on absolute value of directed gradient and sparsity constraints on the image of rain.