The presentation starts with some motivating examples of Inverse Problems before introducing the general setting. We shortly review the most common regularization approaches (Tikhonov, iteration methods) and sketch some recent developments in sparsity and machine learning. Sparsity refers to additional expert information on the desired reconstruction, namely, that is has a finite expension in some predefined basis or frame. In machine learning we focus on 'multi colored' inverse problems, where part of the application can be formulated by a strict analytical framework but some part of the problem needs to modeled by a data driven approach. Those combined problems can be created by data- driven linear low rank approximations or more general black box models. In particular we review deep learning approaches to inverse problems. Finally, machine learning techniques by themselves are often inverse problems. We highlight basis learning techniques and applications to hyperspectral image analysis.