This article studies the nonlinear realisation problem, i.e. the problem of finding the state equations of a nonlinear system from the transfer function representation being easily computable from the higher order input-output differential equation. The realisation in both observer and controller canonical forms is studied. The results demonstrate a clear connection with those from linear theory. In the solution the concept of adjoint polynomials, adjoint transfer function and right factorisation of the transfer function play a key role. Finally, the results are applied for system linearisation up to input-output injection used in the observer design.