In CIMPA 2018, School on Combinatorial Commutative Algebra, we will discuss 6 different themes from Combinatorial Commutative Algebra. The idea of Combinatorial Commutative Algebra is to relate combinatorial objects, like simplicial complexes, graphs, hypergraphs or polytopes to algebraic objects like monomial, binomomial ideals and toric rings. The field benefits from the interplay between properties of combinatorial objects and the corresponding properties of the algebraic object. The binomial edge ideals as well as edge ideals of graphs and their powers as well as their symbolic powers will be considered. Other lectures deal with letterplace ideals and the weak Lefschetz property of algebras defined by monomial ideals. The local cohomology in combinatorial contexts will be another topic. As general reference the books "Combinatorial Commutative Algebra" by Sturmfels and Miller and "Monomial ideals" by Herzog and Hibi are recommended.