The dissertational work is devoted to studying the properties, which are connected with the sensitivity of the dynamical systems to the initial conditions. At the beginning we consider earlier known concepts and results, related to this topic. In the main part the generalization of such results is made and some new statements about the sensitivity of systems are proved. We introduce four Lyapunov numbers, which are the quantitative measures, assigned for the description of extremal and asymptotic characteristics of sensitivity for some system $(X,f)$. Here $X$ is a metric and, as usual, compact space, and $f$ is its self-mapping. A series of equalities and inequalities is proved for the Lyapunov numbers. In partial, it is shown, that in case of a compact space two arbitrary Lyapunov numbers differ at most twice. Also we have shown the equalities between some pairs of these numbers for transitive, minimal and weakly mixing systems. Separately we consider the systems, where a segment stands for the space $X$. For them we also have proved the equalities between the Lyapunov numbers, which not necessarily have place in the general case. Additionally the induced systems, where the points of the space are all segments, which belong to the given one, are studied. It is shown for the systems of such type, that there is always an element, which is stable in the sense of Lyapunov. As a corollary we get, that the mentioned systems can't be sensitive to the initial conditions. Also the systems of more general type, where an arbitrary semigroup of mappings acts instead of a single one, are considered in the dissertation. The concept of Lyapunov numbers, and also the Li-Yorke sensitivity, are transferred to the case of such systems. For this generalized case the equalities and inequalities are obtained. Also we have proved a theorem, which states the following: a weakly mixing system with a compact space, which contains more than one point, and a commutative semigroup of continuous mappings is sensitive in the sense of Li-Yorke. This generalizes the known result for the systems with one mapping.
