Mathematical imaging is a topic that touches upon several areas of mathematics, engineering and computer science, including functional and non-smooth analysis, the theory and numerical analysis of partial differential equations, harmonic, stochastic and statistical analysis, optimisation and machine learning. In this talk we will learn about some of these mathematical problems, about variational models for image analysis and their connection to partial differential equations and about a new paradigm in mathematical imaging using deep neural networks. The talk is furnished with applications to art restoration, forest conservation and cancer research.