A review of matrix scaling and Sinkhorn's normal form for matrices and positive maps

Given a nonnegative matrix A, can you find diagonal matrices D₁,~D₂ that D₁AD₂ is doubly stochastic? The answer to this question is known Sinkhorn's theorem. It has been proved with a wide variety of methods, each a variety of possible generalisations. Recently, generalisations as to positive maps between matrix algebras have become more and more for applications. This text gives a review of over 70 years of scaling. The focus lies on the mathematical landscape surrounding the and its solution as well as the generalisation to positive maps and hardly any nontrivial unpublished results.