  In plane Couette flow, the incompressible fluid between two plane parallel
walls is driven by the motion of those walls. The laminar solution, in which
the streamwise velocity varies linearly in the wall-normal direction, is known
to be linearly stable at all Reynolds numbers ($Re$). Yet, in both experiments
and computations, turbulence is observed for $Re \gtrsim 360$.
  In this article, we show that for certain {\it threshold} perturbations of
the laminar flow, the flow approaches either steady or traveling wave
solutions. These solutions exhibit some aspects of turbulence but are not fully
turbulent even at $Re=4000$. However, these solutions are linearly unstable and
flows that evolve along their unstable directions become fully turbulent. The
solution approached by a threshold perturbation could depend upon the nature of
the perturbation. Surprisingly, the positive eigenvalue that corresponds to one
family of solutions decreases in magnitude with increasing $Re$, with the rate
of decrease given by $Re^{\alpha}$ with $\alpha \approx -0.46$.
