In the study of Markov processes, duality is an important tool used to prove various types of long-time behavior. Nowadays, there exist two predominant approaches to Markov process duality: the algebraic one and the pathwise one. This thesis utilizes the pathwise approach in order to identify new dualities of interacting particle systems and to present previously known dualities within a unified framework. Three classes of pathwise dualities are identified by equipping the state space of an interacting particle system with the additional structure of a monoid, a module over a semiring, and a partially ordered set, respectively. This additional structure then induces a pathwise duality for each interacting particle system that preserves this structure in the sense that its generator can be written using only structure-preserving local maps.