Thelastdecadehasseenanumberofexcitingdevelopmentsattheintersectionofcommutativealgebrawithcombinatorics.Newmethodshaveevolvedoutofaninfluxofideasfromsuchdiverseareasaspolyhedralgeometry,theoreticalphysics,representationtheory,homologicalalgebra,symplecticgeometry,graphtheory,integerprogramming,symboliccomputation,andstatistics.Thepurposeofthisvolumeistoprovideaselfcontainedintroductiontosomeoftheresultingcombinatorialtechniquesfordealingwithpolynomialrings,semigrouprings,anddeterminantalrings.Ourexpositionmainlyconcernscombinatoriallydefinedidealsandtheirquotients,withafocusonnumericalinvariantsandresolutions,especiallyundergradingsmorerefinedthanthestandardintegergrading.