Abstract: High-dimensional data appear naturally in many applications and are an increasingly active research field. We give an overview of recent developments in high-dimensional mean estimation, estimation of higher moments, and linear regression. In the context of stochastic optimization, we prove the existence of estimation-procedures satisfying optimal statistical properties. This is followed by an outlook to future applications in mathematical finance and machine learning. We then move away from mean-estimation and discuss the concept of column randomization. We prove that column randomization strongly regularizes undetermined linear systems which appear e.g. in compressed sensing.