  We analyze the Lagrangian and Hamiltonian formulations of the
Maxwell-Chern-Simons theory defined on a manifold with boundary for two
different sets of boundary equations derived from a variational principle. We
pay special attention to the identification of the infinite chains of boundary
constraints and their resolution. We identify edge observables and their
algebra [which corresponds to the well-known $U(1)$ Kac-Moody algebra]. Without
performing any gauge fixing, and using the Hodge-Morrey theorem, we solve the
Hamilton equations whenever possible. In order to give explicit solutions, we
consider the particular case in which the fields are defined on a $2$-disk.
Finally, we study the Fock quantization of the system and discuss the quantum
edge observables and states.
