Time-resolved fluorescence anisotropy measurements are widely used to probe the dynamics of the various processes that can result in the depolarisation of emission following photoselection by polarised excitation. The most common of these emission depolarising phenomena to be investigated is molecular rotational motion, but such measurements are very useful for determining the kinetics of a host of other processes. In this talk I will review several examples for which we have observed initially unexpectedly complex temporal behaviour of the time-resolved fluorescence anisotropy signal from relatively "simple" systems. In certain circumstances the anisotropy (i) decays on timescales when superficially it might be thought it should remain constant, (ii) shows marked "dip and rise" behaviour in its intensity, or (iii) can change sign as the anisotropy evolves in time. Fundamentally simple processes, including molecular rotational motion, energy migration and excited state photophysics, can cause such behaviour, and will be discussed.