  Using newly developed ${\bf H}(\mathrm{curl}^2)$ conforming elements, we
solve the Maxwell's transmission eigenvalue problem. Both real and complex
eigenvalues are considered. Based on the fixed-point weak formulation with
reasonable assumptions, the optimal error estimates for numerical eigenvalues
and eigenfunctions (in the ${\bf H}(\mathrm{curl}^2)$-norm and ${\bf
H}(\mathrm{curl})$-semi-norm) are established.
