This talk explores the developments in unbalanced optimal transport (OT) in a constrained setting. Beginning with a fundamental introduction to unbalanced OT, we discuss the Wasserstein-Fisher-Rao distance, which offers a comprehensive perspective on the space of measures. As we venture deeper, the conversation expands from the basic constraints, realizing area measures of convex sets to the complexities of more generalized constraints in OT. A significant portion of the talk is dedicated to proving the existence of minimal energy paths, focusing on the Fenchel-Rockafellar theorem and its implications. Moreover, we discuss the version of Otto's picture in our setting, providing a geometric perspective of the theory. This talk promises a profound understanding of unbalanced, constrained OT, bridging foundational concepts with contemporary advancements.