Finite element spaces on a tetrahedron are constructed for
div
div
\operatorname {div}\operatorname {div}
-conforming symmetric tensors in three dimensions. The key tools of the construction are the decomposition of polynomial tensor spaces and the characterization of the trace operators. First, the
div
div
\operatorname {div}\operatorname {div}
Hilbert complex and its corresponding polynomial complexes are presented. Several decompositions of polynomial vector and tensor spaces are derived from the polynomial complexes. Second, traces for the
div
div
\operatorname {div}\operatorname {div}
operator are characterized through a Green’s identity. Besides the normal-normal component, another trace involving combination of first order derivatives of the tensor is continuous across the face. Due to the smoothness of polynomials, the symmetric tensor element is also continuous at vertices, and on the plane orthogonal to each edge. Besides, a finite element for
s
y
m
c
u
r
l
symcurl
-conforming trace-free tensors is constructed following the same approach. Putting all together, a finite element
div
div
\operatorname {div}\operatorname {div}
complex, as well as the bubble functions complex, in three dimensions is established.