  A consistent approach to the description of integral coordinate invariant
functionals of the metric on manifolds ${\cal M}_{\alpha}$ with conical defects
(or singularities) of the topology $C_{\alpha}\times\Sigma$ is developed.
According to the proposed prescription ${\cal M}_{\alpha}$ are considered as
limits of the converging sequences of smooth spaces. This enables one to give a
strict mathematical meaning to a number of invariant integral quantities on
${\cal M}_{\alpha}$ and make use of them in applications. In particular, an
explicit representation for the Euler numbers and Hirtzebruch signature in the
presence of conical singularities is found. Also, higher dimensional Lovelock
gravity on ${\cal M}_{\alpha}$ is shown to be well-defined and the
gravitational action in this theory is evaluated. Other series of applications
is related to computation of black hole entropy in the higher derivative
gravity and in quantum 2-dimensional models. This is based on its direct
statistical-mechanical derivation in the Gibbons-Hawking approach, generalized
to the singular manifolds ${\cal M}_{\alpha}$, and gives the same results as in
the other methods.
