Abstract <p> The paper is devoted to the study of stability of solutions of a new class of linearfunctionally integral Itô equations, which contains many classical equations, e.g.,differential equations of integer and fractional order with and without stochastic perturbationsthem, as well as some less known and understudied types of equations that have been introducedinto scientific circulation recently. The connection between different types of stability of solutionsof these equations and belonging of their solutions to the corresponding spaces of randomprocesses. Using this connection sufficient conditions of stability of solutions with respect to initialdata in terms of parameters of these equations. The notion of admissibility of pairs of spaces forthe above mentioned the notion of admissibility of pairs of spaces for the above mentionedequations and the relationship between admissibility of pairs of spaces and stability with respectto the initial function.</p>