Sensory information about the outside world is encoded by neurons in sequences of discrete, identical pulses termed action potentials or spikes. There is persistent controversy about the extent to which the precise timing of these spikes is relevant to the function of the brain. We revisit this issue, using the motion-sensitive neurons of the fly visual system as a test case. Our experimental methods allow us to deliver more nearly natural visual stimuli, comparable to those which flies encounter in free, acrobatic flight. New mathematical methods allow us to draw more reliable conclusions about the information content of neural responses even when the set of possible responses is very large. We find that significant amounts of visual information are represented by details of the spike train at millisecond and sub-millisecond precision, even though the sensory input has a correlation time of approximately 55 ms, different patterns of spike timing represent distinct motion trajectories, and the absolute timing of spikes points to particular features of these trajectories with high precision. Finally, the efficiency of our entropy estimator makes it possible to uncover features of neural coding relevant for natural visual stimuli: first, the systems information transmission rate varies with natural fluctuations in light intensity, resulting from varying cloud cover, such that marginal increases in information rate thus occur even when the individual photoreceptors are counting on the order of one million photons per second. Secondly, we see that the system exploits the relatively slow dynamics of the stimulus to remove coding redundancy and so generate a more efficient neural code.