We consider the observability of systems of the form \(\dot x=Ax+Nx,y=Fx\), where A is a linear operator and N and F are nonlinear. We show that if the system is linearized about an equilibrium point \(x_ e\) and the linearized system is continuously initially observable, then the nonlinear system is continuously initially observable in some neighbourhood of \(x_ e\). We then look at conditions under which solutions of the nonlinear system can be extended for all time and consider the problem of stabilizing the system by feedback controls such that the solutions are eventually in the observability neighbourhood of \(x_ e\). Finally, we aply these ideas to two systems: a wave equation and a diffusion equation with nonlinear perturbations and nonlinear observations.