Summary: A general theory which can be used to derive bounds on solutions to the Navier-Stokes equations is presented. The behaviour of the resolvent of the linear operator in the unstable half-plane is used to bound the energy growth of the full nonlinear problem. Plane Couette flow is used as an example. The norm of the resolvent in plane Couette flow in the unstable half-plane is proportional to the square of the Reynolds number \((R)\). This is now used to predict the asymptotic behaviour of the threshold amplitude below which all disturbances eventually decay. A lower bound is found to be \(R^{-21/4}\). Examples, obtained through direct numerical simulation, give an upper bound on the threshold curve, and predict a threshold of \(R^{-1}\). The discrepancy is discussed in the light of a model problem.