Ricci flows are used as a powerful tool to address several problems of information technology, engineering, medical science, and other allied areas. A self-similar solution of the Ricci flow on a Riemannian manifold is named as the Ricci soliton. The Ricci soliton becomes an almost Ricci soliton if we think of the soliton constant as a smooth function in the Ricci soliton equation. This chapter explores the properties of almost Ricci solitons within the framework of Riemannian concircular structure manifolds (briefly, \((RCS)_n\) -manifolds). We establish the conditions for which the \((RCS)_n\) -manifolds to be quasi-Einstein manifolds, and the solitons are expanding, shrinking, and steady. We also provide the restrictions for the soliton function of almost Ricci solitons to be harmonic, strictly super-harmonic, and strictly subharmonic. The existence of projectively semisymmetric \((RCS)_n\) -manifolds is ensured. Some geometrical properties of \((RCS)_n\) -manifolds satisfying \(Q\cdot \mathcal {P}=0\) are investigated, and the definition of extended Ricci recurrent manifolds is encoded.