Recently, Davidson and I introduced a new framework for noncommutative convexity and noncommutative function theory, along with a corresponding noncommutative Choquet theory that generalizes much of classical Choquet theory. A key point is that the category of compact noncommutative sets is dual to the category of operator systems. In recent work with Kim and Manor, we extended this duality to the category of (potentially) non-unital operator systems in the sense of Werner and Connes-van Suijlekom. I will motivate and discuss these developments, as well as some applications to C*-algebras and noncommutative dynamics.