Monomial ideals are at the intersection between commutative algebra, combinatorics, and algebraic geometry, being in the center of many important problems in polynomial rings. Their study developed extensively especially in the last decades when it became a standard technique to get information about the algebraic and homological invariants of polynomial ideals by passing to initial monomial ideals. By using a standard procedure, one may also use specific combinatorial techniques to study invariants of the so called squarefree monomial ideals. All these new developments led to a spectacular...