A classical problem in complex function theory is to identify the unique smallest boundary associated with a subalgebra of continuous functions on a compact Hausdorff space. This boundary is typically called the Choquet boundary of the subalgebra and several distinct interpretations of the Choquet boundary are available. We investigate non-commutative interpretations of these different characterizations for unital subalgebras of bounded linear operators. In the process, we identify the notion of a non-commutative boundary and consequently obtain a characterization of minimality for the non-commutative Choquet boundary. In particular, the non-commutative Choquet boundary is not always a minimal boundary. Nonetheless, natural examples from multivariate operator theory demonstrate that the non-commutative Choquet boundary can be minimal in special cases. This is joint work with Raphaël Clouâtre.