Non-unital operator systems are norm closed subspaces of B(H) that are closed under the involution map $x \mapsto x^*$. For example, C*-algebras are examples of non-unital operator systems. Much like Gelfand duality, a result of Kadison from the 80s shows that operator systems generated by commuting elements are categorically dual to a class of geometric structures, namely compact convex sets. Unlike the C*-theory, a remarkable result due to Webster-Winkler and Davidson-Kennedy shows that Kadison's duality theorem readily generalizes to the non-commutative unital setting as well. In our talk, we discuss Kadison’s original duality as well as the nc convex duality for non-commutative operator systems due to myself, Matt Kennedy, and Nick Manor. Finally, we will have a discussion of some results on the side of operator systems that this illuminates.