  We prove $hp$-optimal error estimates for interior penalty discontinuous
Galerkin methods (IPDG) for the biharmonic problem with homogeneous essential
boundary conditions. We consider tensor product-type meshes in two and three
dimensions, and triangular meshes in two dimensions. An essential ingredient in
the analysis is the construction of a global $H^2$ piecewise polynomial
approximants with $hp$-optimal approximation properties over the given meshes.
The $hp$-optimality is also discussed for $\mathcal C^0$-IPDG in two and three
dimensions, and the stream formulation of the Stokes problem in two dimensions.
Numerical experiments validate the theoretical predictions and reveal that
$p$-suboptimality occurs in presence of singular essential boundary conditions.
