  The file contains PhD Dissertation by Oskar Jakub Szyma\'nski. This work ends
his study at Doctoral School of Exact and Natural Sciences at Jagiellonian
University where the Author has attended in years 2019-2024. The subject of the
Thesis is the stability theory developed for matrix polynomials. The main
concept described in the Dissertation is hyperstability with respect to an open
or closed subset of the complex plane. The notion of hyperstability is a
contribution of the Author of the Thesis. Disseration is devoted mostly to
regular polynomials, singular polynomials play marginal role in the Thesis. The
Author of the Thesis has discovered several interesting results extending
classical theorems from complex analysis. These are for example: multi-variable
Gauss-Lucas theorem for matrix polynomials and Sz\'asz-type inequalities for
matrix norms such as the matrix two-norm and the Frobenius norm.
