  A smooth hypersurface over a finite field $\mathbb{F}_q$ is called Frobenius
nonclassical if the image of every geometric point under the $q$-th Frobenius
endomorphism remains in the unique hyperplane tangent to the point. In this
paper, we establish sharp lower and upper bounds for the degrees of such
hypersurfaces, give characterizations for those achieving the maximal degrees,
and prove in the surface case that they are Hermitian when their degrees attain
the minimum. We also prove that the set of $\mathbb{F}_q$-rational points on a
Frobenius nonclassical hypersurface form a blocking set with respect to lines,
which indicates the existence of many $\mathbb{F}_q$-points.
