**Abstract:** Let $X_1,\ldots,X_N$ denote $N$ independent, symmetric L\'evy processes on $\mathbf{R}^d$. The corresponding \emph{additive L\'evy process} is defined as the following $N$-parameter random field on $\mathbf{R}^d$: \begin{equation} X(\mathbf{t}) :=X_1(t_1) + \cdots + X_N(t_N)\qquad(\mathbf{t}\in \mathbf{R}^N_+). \end{equation} Khoshnevisan and Xiao (2002) have found a necessary and sufficient condition for the zero-set $X^{-1}(\{0\})$ of $X$ to be non-trivial with positive probability. They also provide bounds for the Hausdorff dimension of $X^{-1}(\{0\})$ which hold with positive probability in the case that $X^{-1}(\{0\})$ can be non-void. Here, we prove that the Hausdorff dimension of $X^{-1}(\{0\})$ is a constant almost surely on the event $\{ X^{-1}(\{0\})\neq\emptyset \}$. Moreover, we derive a formula for the said constant. This portion of our work extend the one-parameter formulas of Horowitz (1968) and Hawkes (1974). More generally, we prove that for every non-random Borel set $F$ in $(0\,,\infty)^N$, the Hausdorff dimension of $X^{-1}(\{0\})\cap F$ is a constant almost surely on the event $\{ X^{-1}(\{0\})\cap F\neq\emptyset\}$. This constant is computed explicitly in many cases. (Joint work with Narn-Rueih Shieh and Yimin Xiao)