We examine the ability of a target zone to stabilize exchange rates in the presence of two stylised forms of market inefficiency - stochastic bubbles and "fads". We show how the usual saddlepath phase diagram is modified in the presence of bubbles (both where fundamentals are deterministic and where they are stochastic), and how a credible policy to defend a target zone prevents the emergence of any such bubbles. In the case of fads, we suppose that the source of shocks driving the exchange rate from its long-run equilibrium level is the fluctuating sentiment of "noise traders". The presence of "smart money" in the market exerts a stabilizing influence upon the exchange rate even in the absence of a target zone. In the presence of a fully credible target zone, where the authorities can take the necessary actions to check the swings in market sentiment, we use results on regulated Brownian motion to characterise the exchange rate trajectory, and demonstrate that the stabilizing influence of "smart money" is increased.