A new
H
(
div
div
)
H(\operatorname {div}\operatorname {div})
-conforming finite element is presented, which avoids the need for supersmoothness by redistributing the degrees of freedom to edges and faces. This leads to a hybridizable mixed method with superconvergence for the biharmonic equation. Moreover, new finite element divdiv complexes are established. Finally, new weak Galerkin and
C
0
C^0
discontinuous Galerkin methods for the biharmonic equation are derived.