Optimal transport has a long standing history in different branches of science, originating from physics and logistics. Since its emergence as a rigorous mathematical concept in the last century it has influenced and shaped various areas within mathematics but it also has been proven to a be a remarkably rich and fruitful concept for various related disciplines, such as economics and more recently computer science, machine learning and statistics. In this talk we discuss classical and more recent developments in statistical data analysis based on empirical optimal transport (EOT). Our mathematical fundament are limit laws and risk bounds for EOT plans and distances on finite and discrete spaces. Proofs are based on a combination of sensitivity analysis from convex optimization and discrete empirical process theory. Theory will be used for statistical inference, fast simulation, and for fast randomized computation of optimal transport in large scale data applications at pre-specified computational cost. EOT based data analysis is illustrated in various computer experiments and on biological data from super-resolution cell microscopy.