  In 1998, Georgia Benkart and Tom Roby introduced the down-up algebra
$\mathcal A$. The algebra $\mathcal A$ is associative, noncommutative, and
infinite-dimensional. It is defined by two generators $A,B$ and two relations
called the down-up relations. In the present paper, we introduce the $\mathbb
Z_3$-symmetric down-up algebra $\mathbb A$. We define $\mathbb A$ by generators
and relations. There are three generators $A,B,C$ and any two of these satisfy
the down-up relations. We describe how $\mathbb A$ is related to some familiar
algebras in the literature, such as the Weyl algebra, the Lie algebras
$\mathfrak{sl}_2$ and $\mathfrak{sl}_3$, the $\mathfrak{sl}_3$ loop algebra,
the Kac-Moody Lie algebra $A^{(1)}_2$, the $q$-Weyl algebra, the quantized
enveloping algebra $U_q(\mathfrak{sl}_2)$, and the quantized enveloping algebra
$U_q (A^{(1)}_2)$. We give some open problems and conjectures.
