  I discuss possible consequences of A. D. Sakharov's hypothesis of
cosmological transitions with changes in the signature of the metric, based on
the path integral approach. This hypothesis raises a number of mathematical and
philosophical questions. Mathematical questions concern the definition of the
path integral to include integration over spacetime regions with different
signatures of the metric. One possible way to describe the changes in the
signature is to admit time and space coordinates to be purely imaginary. It may
look like a generalization of what we have in the case of pseudo-Riemannian
manifolds with a non-trivial topology. The signature in these regions can be
fixed by special gauge conditions on components of the metric tensor. The
problem is what boundary conditions should be imposed on the boundaries of
these regions and how they should be taken into account in the definition of
the path integral. The philosophical question is what distinguishes the time
coordinate among other coordinates but the sign of the corresponding principal
value of the metric tensor. In particular, I try to speculate how the existence
of the regions with different signature can affect the evolution of the
Universe.
