In this thesis, we have investigated conformal symmetries of the Ricci tensor, also known as conformal Ricci collineations (CRCs), for certain physically important spacetimes including kantowski-Sachs spacetimes, static spacetimes with maximal symmetric transverse spaces, non-static spherically symmetric spacetimes and locally rotationally symmetric Bianchi type I and V spacetimes. For each of these spacetimes, the CRC equations are solved in degenerate as well as non-degenerate cases. When the Ricci tensor is degenerate, it is observed that for all the above mentioned spacetimes, the Lie algebra of CRCs is in nite-dimensional. For non-degenerate Ricci tensor, it is shown that the spacetimes under consideration always admit anite-dimensional Lie algebra of CRCs. For Kantowski-Sachs and locally rotationally symmetric Bianchi type V metrics, we obtain 15-dimensional Lie algebras of CRCs, which is the maximum dimension of conformal algebra for a spacetime. In case of static spacetimes with maximal symmetric transverse spaces, the dimension of Lie algebra of CRCs turned out to be 6, 7 or 15. Similarly, it is observed that non-static spherically symmetric spacetimes may possess 5, 6 or 15 CRCs for non-degenerate Ricci tensor. Finally, the dimension of Lie algebra of CRCs for locally rotationally symmetric Bianchi type I spacetimes is shown to be 7- or 15-dimensional. For all the above mentioned spacetimes, the CRCs are found subject to some highly non-linear di erential constraints. In order to show that the classes of CRCs are non-empty, some examples of exact form of the corresponding metric satisfying these constraints are provided.