A decade ago, Katsoulis and Ramsey began a program for studying non-self-adjoint dynamical systems and their enveloping C*-algebras. Within their program, the central problem was to determine the C*-envelope of the non-self-adjoint reduced and full crossed product operator algebras. This is of paramount importance for various classification statements as, for one, this would provide answers to the 17 year-old reduced and full Hao-Ng isomorphism problems. In past work with Adam Dor-On, we gave a complete positive answer to the reduced case. In forthcoming work, we provide a corresponding answer to the much more mysterious setting of full crossed products, with a complete answer to the full Hao-Ng isomorphism problem.