We review recent work on the transition to turbulence in shear flows, in particular plane Couette flow. In the linearized system the non-normality of the linear operator gives rise to non-orthogonal eigenvectors and to a significant amplification of noise. We present a typical result for a Fokker-Planck equation with non-normal relaxation matrix. As the driving becomes stronger, further stationary states are born in a saddle node bifurcation. This is observed in a two degree of freedom phenomenological model, in the a few mode approximation to a simple shear flow and in full numerical studies of plane Couette flow. The stationary states give rise to a fractal border between decaying and turbulent states.