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Distribution system planning and operation has seen many structural changes due to the increased participation of consumers in the energy market and the adaptation of new technologies such as distributed energy resources (DERs), electric vehicles (EV) and local energy storage systems (ESSs). Despite the convenience of such technologies and the gradual drop in their prices, new technical challenges (e.g., excessive power losses) have emerged at the system level. Over the past few decades, power loss minimization in distribution systems has gained popularity and the need for loss sensitivity analysis (LSA) frameworks has become a necessity for its successful implementation. Existing work on LSA mostly focuses on system planning aspects through DER optimal placement and sizing. However, enabling LSA-based system operational applications is a vital step toward the successful transition to modern distribution systems (MDSs). Therefore, this paper presents a comprehensive overview on the state of the art in LSA for MDSs. First, the theoretical formulations of existing LSA methods are summarized. Then, the applications of LSA in distribution systems are highlighted. Finally, based on the analysis of literature, open research gaps and future research pathways are discussed.