  We show in this article how the usual hamiltonian formalism of General
Relativity should be modified in order to allow the inclusion of the Euclidean
classical solutions of Einstein's equations. We study the effect that the
dynamical change of signature has on the superspace and we prove that it
induces a passage of the signature of the supermetric from ($-+++++$) to
($+-----$). Next, all these features are more particularly studied on the
example of minisuperspaces. Finally, we consider the problem of quantization of
the Euclidean solutions. The consequences of different choices of boundary
conditions are examined.
