We hybridize the methods of finite element exterior calculus for the Hodge–Laplace problem on differential
k
k
-forms in
R
n
\mathbb {R}^n
. In the cases
k
=
0
k=0
and
k
=
n
k=n
, we recover well-known primal and mixed hybrid methods for the scalar Poisson equation, while for
0
>
k
>
n
0>k>n
, we obtain new hybrid finite element methods, including methods for the vector Poisson equation in
n
=
2
n=2
and
n
=
3
n=3
dimensions. We also generalize Stenberg postprocessing [RAIRO Modél. Math. Anal. Numér. 25 (1991), pp. 151–167] from
k
=
n
k=n
to arbitrary
k
k
, proving new superconvergence estimates. Finally, we discuss how this hybridization framework may be extended to include nonconforming and hybridizable discontinuous Galerkin methods.