The fact that distillation columns, even in the ideal binary case, may display multiple steady states and unstable operating points has only recently been recognized. This article addresses some implications of these phenomena for the operation and control of distillation columns. Under manual operation, the multiplicity and instability will result in inability to reach separations corresponding to unstable operating points and may furthermore cause abrupt changes and hysteresis in operating conditions. It is shown that an unstable operating point may be stabilized by feedback control of a single product composition or tray temperature (one-point control). The steady-state multiplicity does, in this case, not represent any severe limitation in operation, but if the control is not sufficiently tight, the column may settle in sustained oscillations (stable limit cycle). Finally, the impact of open-loop instability on the achievable closed-loop performance with both product compositions under feedback control is discussed.