  One of the most efficient methods for the evaluation of the vacuum
expectation values for physical observables in the Casimir effect is based on
using the Abel-Plana summation formula. This enables to derive the renormalized
quantities in a manifestly cutoff independent way and to present them in the
form of strongly convergent integrals. However, applications of the Abel-Plana
formula, in its usual form, are restricted by simple geometries when the
eigenmodes have a simple dependence on quantum numbers. The author generalized
the Abel-Plana formula which essentially enlarges its application range. Based
on this generalization, formulae have been obtained for various types of series
over the zeros of combinations of Bessel functions and for integrals involving
these functions. It has been shown that these results generalize the special
cases existing in literature. Further, the derived summation formulae have been
used to summarize series arising in the direct mode summation approach to the
Casimir effect for spherically and cylindrically symmetric boundaries, for
boundaries moving with uniform proper acceleration, and in various braneworld
scenarios. This allows to extract from the vacuum expectation values of local
physical observables the parts corresponding to the geometry without boundaries
and to present the boundary-induced parts in terms of integrals strongly
convergent for the points away from the boundaries. As a result, the
renormalization procedure for these observables is reduced to the corresponding
procedure for bulks without boundaries. The present paper reviews these
results. We also aim to collect the results on vacuum expectation values for
local physical observables such as the field square and the energy-momentum
tensor in manifolds with boundaries for various bulk and boundary geometries.
