A K-coloring of a matrix is a partition of its column set into K subsets so that each row satisfies a specified property.
This property is often equivalent to the integrality of a polytope.
We survey the major results in the area, starting from the classical bicoloring theorems of Ghouila-Houri and Berge on bicoloring of Totally Unimodular and balanced matrices and then covering more recent results obtained by Conforti, Cornuejols and Zambelli on colorings of balanced 0,±1 matrices and k-balanced matrices. We will link these results to the integrality of some (generalized) packing and covering polytopes.