  This lectures notes consists of four lectures. The first lecture discusses
questions around Hilbert-Arnold Problem which is naturally arises from
Quantitative Hilbert 16-th problem. In the second lecture we outline author's
solution of a weak form of Local Hilbert-Arnold Problem. This solution provides
an independent proof of Ilyashenko-Yakovenko Finiteness Theorem. The third
lecture discusses question of existence of a_P-stratification of Thom and
presents a simple geometric proof of existence of such a stratification for
polynomial functions, which was originally proven by Hironaka. The forth
lecture gives application of Grigoriev-Yakovenko's construction to the problem
of growth of the number of periodic points and the problem of bifurcation of
spacial polycycles. The latter problem naturally generalizes Local
Hilbert-Arnold Problem.
