  In this paper, we introduce the notion of DG Sklyanin algebras, which are
connected cochain DG algebras whose underlying graded algebras are Sklyanin
algebras. Let $\mathcal{A}$ be a $3$-dimensional DG Sklyanin algebra with
$\mathcal{A}^{\#}=S_{a,b,c}$, where $(a,b,c)\in \Bbb{P}_k^2-\mathfrak{D}$ and
$$\mathfrak{D}=\{(1,0,0), (0,1,0),(0,0,1)\}\sqcup\{(a,b,c)|a^3=b^3=c^3\}.$$ We
systematically study its differential structures and various homological
properties. Especially, we figure out the conditions for $\mathcal{A}$ to be
Calabi-Yau, Koszul, Gorenstein and homologically smooth, respectively.
