Abstarct: In the first part of this thesis, we reformulate Einstein’s theory of gravity, isolating the conformal degree of freedom in a covariant way. This is done by introducing a physical metric defined in terms of an auxiliary metric and a scalar field appearing through its first derivatives. The resulting equations of motion split into a traceless equation obtained through variation with respect to the auxiliary metric and an additional differential equation for the trace part. As a result, the conformal degree of freedom becomes dynamical even in the absence of matter. We show that this extra degree of freedom can mimic cold dark matter[1]. After this, by examining the cosmological perturbations up to the second order for the mimetic single field model, we see that we do not have the dynamic state for the scalar field. Then, in order to remdy this problem, we study the cosmological disturbances for the mimetic model with the arbitrary function of the higher order derivative. We also calculate the second-order action and show that a new problem arises, both the kinetic term and the gradient term (potential) have the wrong sign, which leads to problems such as ghosting and gradient instability [2]. In the last part of this thesis, we extend the mimetic gravity to the multi-field setup with a curved field space manifold. The multi-field mimetic scenario is realized via the singular limit of the conformal transformation between the auxiliary and the physical metrics. We look for the cosmological implications of the setup where it is shown that at the background level the mimetic energy density mimics the roles of dark matter. At the perturbation level, the scalar field perturbations are decomposed into the tangential and normal components with respect to the background field space trajectory. The adiabatic perturbation tangential to the background trajectory is frozen while the entropy mode perpendicular to the background trajectory propagates with the speed of unity. Whether or not the entropy perturbation is healthy directly depends on the signature of the field-space metric [3].