  A tensor $\mathcal T\in \mathbb T(\mathbb C^n,m+1)$, the space of tensors of
order $m+1$ and dimension $n$ with complex entries, has $nm^{n-1}$ eigenvalues
(counted with algebraic multiplicities). The inverse eigenvalue problem for
tensors is a generalization of that for matrices. Namely, given a multiset
$S\in \mathbb C^{nm^{n-1}}/\mathfrak{S}(nm^{n-1})$ of total multiplicity
$nm^{n-1}$, is there a tensor in $\mathbb T(\mathbb C^n,m+1)$ such that the
multiset of eigenvalues of $\mathcal{T}$ is exact $S$? The solvability of the
inverse eigenvalue problem for tensors is studied in this paper. With tools
from algebraic geometry, it is proved that the necessary and sufficient
condition for this inverse problem to be generically solvable is $m=1,\
\text{or }n=2,\ \text{or }(n,m)=(3,2),\ (4,2),\ (3,3)$.
