  In this paper, we first construct the $H^2$(curl)-conforming finite elements
both on a rectangle and a triangle. They possess some fascinating properties
which have been proven by a rigorous theoretical analysis. Then we apply the
elements to construct a finite element space for discretizing quad-curl
problems. Convergence orders $O(h^k)$ in the $H$(curl) norm and $O(h^{k-1})$ in
the $H^2$(curl) norm are established. Numerical experiments are provided to
confirm our theoretical results.
