In this chapter, we establish the boundedness of (in general) sublinear integral operators and their commutators in weighted classical and grand Morrey spaces defined over a space of homogeneous type (SHT), under the assumption that the weights satisfy the Muckenhoupt condition. These operators include, for example, the Hardy–Littlewood and fractional maximal operators, Calderón–Zygmund operators, potential-type operators, and others. Applications are also provided for estimates for hypoelliptic operators in weighted Morrey spaces defined on nilpotent Lie groups. Moreover, we obtain boundedness criteria for the Hardy–Littlewood maximal operator and the Riesz transforms in the weighted grand Morrey spaces \(M^{p),q,\varphi }_w\) , expressed in terms of Muckenhoupt weights.