  The Casimir energies and pressures for a massless scalar field associated
with $\delta$-function potentials in 1+1 and 3+1 dimensions are calculated. For
parallel plane surfaces, the results are finite, coincide with the pressures
associated with Dirichlet planes in the limit of strong coupling, and for weak
coupling do not possess a power-series expansion in 1+1 dimension. The relation
between Casimir energies and Casimir pressures is clarified,and the former are
shown to involve surface terms. The Casimir energy for a $\delta$-function
spherical shell in 3+1 dimensions has an expression that reduces to the
familiar result for a Dirichlet shell in the strong-coupling limit. However,
the Casimir energy for finite coupling possesses a logarithmic divergence first
appearing in third order in the weak-coupling expansion, which seems
unremovable. The corresponding energies and pressures for a derivative of a
$\delta$-function potential for the same spherical geometry generalizes the TM
contributions of electrodynamics. Cancellation of divergences can occur between
the TE ($\delta$-function) and TM (derivative of $\delta$-function) Casimir
energies. These results clarify recent discussions in the literature.
