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A graph $G$ with $p$ vertices and $q$ edges, vertex set $V\left(G\right)$ and edge set $E\left(G\right)$, is said to be super vertex-graceful (in short SVG), if there exists a function pair $(f,f^{+})$ where $f$ is a bijection from $V\left(G\right)$ onto $P$, $f^{+}$ is a bijection from $E\left(G\right)$ onto $Q$, $f^{+}\left((u,v)\right)=f\left(u\right)+f\left(v\right)$ for any $(u,v)\in E\left(G\right)$, $$Q=\left\{\begin{array}{cc}\{\pm 1,\cdots ,\pm \frac{1}{2}q\},\hfill & \text{if}q\text{is}\text{even,}\hfill \\ \{0,\pm 1,\cdots ,\pm \frac{1}{2}(q-1)\},\hfill & \text{if}q\text{is}\text{odd,}\hfill \end{array}\right.$$ and $$P=\left\{\begin{array}{cc}\{\pm 1,\cdots ,\pm \frac{1}{2}p\},\hfill & \text{if}p\text{is}\text{even,}\hfill \\ \{0,\pm 1,\cdots ,\pm \frac{1}{2}(p-1)\},\hfill & \text{if}p\text{is}\text{odd.}\hfill \end{array}\right.$$ We determine here families of unicyclic graphs that are super vertex-graceful.