A polynomial in the ring Q[x1, . . . , xn] is called symmetric if it is invariant under any permutation of the variables x1, . . . , xn. We will overview various properties of the sub-ring of symmetric polynomials, the ‘invariant ideal ’Ingenerated by symmetric polynomials with vanishing constant term, and the quotient Rn of the polynomial ringQ[x1, . . . , xn] by In. The coinvariant algebra Rn has algebraic properties which are deeply tied to the combinatorics of permutations in the symmetric group Sn. Time permitting, we will discuss a new generalization Rn,k of the coinvariant algebra whose properties are similarly related to the combinatorics of ordered set partitions.