  In the context of integrable field theory with boundary, the integrable
non-linear sigma models in two dimensions, for example, the $O(N)$, the
principal chiral, the ${\rm CP}^{N-1}$ and the complex Grassmannian sigma
models are discussed on a half plane. In contrast to the well known cases of
sine-Gordon, non-linear Schr\"odinger and affine Toda field theories, these
non-linear sigma models in two dimensions are not classically integrable if
restricted on a half plane. It is shown that the infinite set of non-local
charges characterising the integrability on the whole plane is not conserved
for the free (Neumann) boundary condition. If we require that these non-local
charges to be conserved, then the solutions become trivial.
