The paper represents a survey of recent results obtained by the author. These results concern the problem of the existence of selectors wich inherit the properties of a multivalued mapping of one variable with compact values in a metric space. The properties considered are the boundedness of Jordan, essential or generalized variation, Lipschitz or absolute continuity. Obtained results are based on the generalization of the Helly selection principle. Applications to functional inclusions and embeddings including variable set-valued operators in the right-hand side are considered.