  We develop and apply an analytic method to predict peak counts in
weak-lensing surveys. It is based on the theory of Gaussian random fields and
suitable to quantify the level of spurious detections caused by chance
projections of large-scale structures as well as the shape and shot noise
contributed by the background galaxies. We compare our method to peak counts
obtained from numerical ray-tracing simulations and find good agreement at the
expected level. The number of peak detections depends substantially on the
shape and size of the filter applied to the gravitational shear field. Our main
results are that weak-lensing peak counts are dominated by spurious detections
up to signal-to-noise ratios of 3--5 and that most filters yield only a few
detections per square degree above this level, while a filter optimised for
suppressing large-scale structure noise returns up to an order of magnitude
more.
