Project: Relatively free systems play a crucial role in studying varieties of algebras. Dealing with such problems, it is important to know the structure of free objects in the variety. The variety theory of Loday’s structures is not so well developed. Partially it is due our lack of knowledge on relatively free Loday’s algebras. This project is devoted to studying relatively free trioids and relatively free generalized dimonoids. The main focus is to construct new free algebras in the varieties of trioids and generalized dimonoids, and investigate the properties of the constructed free algebras. The results of the project can be used by the specialists in different areas of algebra, in particular, in universal algebra andsemigroup theory.