  We make an attempt to map the integrable boundary conditions for 2
dimensional non-linear O(N) $\sigma$-models. We do it at various levels:
classically, by demanding the existence of infinitely many conserved local
charges and also by constructing the double row transfer matrix from the Lax
connection, which leads to the spectral curve formulation of the problem; at
the quantum level, we describe the solutions of the boundary Yang-Baxter
equation and derive the Bethe-Yang equations. We then show how to connect the
thermodynamic limit of the boundary Bethe-Yang equations to the spectral curve.
