Among regularization techniques for optimal transport, entropic regularization has played a pivotal rule. The main reason may be its computational simplicity: the Sinkhorn-Knopp iteration can be implemented in two- or even one line ad enjoys a linear convergence rate. However, some care is needed to calculate optimizer for small regularization parameters and convergence can be quite slow for badly behaved data. Faster algorithms, e.g. Newton methods, are hard to analyze and tend to be unstable in practice. Moreover, the continuous theory is intricate in this case and takes place in Orlicz-Luxemburg spaces (as we will illustrate in this talk). After sketching parts of the continuous theory for entropic regularization, we will analyze a different regularizer, namely a simple quadratic penalty. First our focus lies on the continuous case where it is still quite challenging to show existence of suitable solutions for the dual problem. Then we will derive different numerical methods for the discrete problem which include a globally convergent Newton method which converges very fast to high accuracy even for fairly small regularization parameters. The talk is based on joint work with Christoph Brauer, Christian Clason, Paul Manns, Christian Meyer, and Benedikt Wirth.