In this review we consider in detail different theoretical topics associated with interaction in the dark sector. We study linear and non-linear interactions which depend on the dark matter and dark energy densities. We consider a number of different models (including the holographic dark energy and dark energy in a fractal universe) with interacting dark energy (DE) and dark matter (DM), have done a thorough analysis of these models. The main task of this review was not only to give an idea about the modern set of different models of dark energy, but to show how much can be diverse dynamics of the universe in these models. We find that the dynamics of a Universe that contains interaction in the dark sector can differ significantly from the Standard Cosmological Model (SCM).

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Abstract. Agegraphic dark energy, has been recently proposed, based on the so-called Karolyhazy uncertainty relation, which arises from quantum mechanics together with general relativity. In the first part of the article we study the original agegraphic dark energy model by including the interaction between agegraphic dark energy and pressureless (dark) matter. The phase space analysis was made and the critical points were found, one of which is the attractor corresponding to an accelerated expanding Universe.Recent observations of near supernova show that the acceleration of Universe decreases. This phenomenon is called the transient acceleration. In the second part of Article we consider the 3-component Universe composed of a scalar field, interacting with the dark matter on the agegraphic dark energy background. We show that the transient acceleration appears in frame of such a model. The obtained results agree with the observations.

We investigate interacting dark energy models in the framework of fractal cosmology.We discuss a fractal FRW universe filled with the dark energy and dark matter which interact with each other. We obtain the equation for the relative density of dark matter and dark energy and the deceleration parameter. This model demonstrates new types of evolution, which are not common to cosmological models with this type of interaction.

We consider three cosmological models with linear interaction between the dark components and obtain restrictions on the coupling constant in terms of the cosmographic parameters. It enables us to find constraints on the coupling constant directly based on observational data and to restrict number of numerous models describing interaction in the dark sector.