We have studied the variation of astronomical refraction near the horizon. We have collected 144 measurements of refraction from seven sites by three techniques and have found that the variation of refraction on the horizon is substantially larger than has previously been realized. The rms deviation of our observations is 0?16, while the individual measurements range from 0?234 to 1?678. At the 95% confidence level the total refraction should vary over a range of 0?64. This surprising result has five applications: First, the time of sunrise can only be predicted with an accuracy of 4 min, despite all the extreme accuracy of modern positional astronomy. Second, during a low-altitude solar eclipse (such as is the 1990 eclipse viewed from Finland), the size and shape of the edge of the umbra will vary in an unpredictable manner by perhaps several kilometers, so that attempts to measure the diameter of the Sun may have an unexpected accuracy limit. Contact times may vary by typically 0.1 sec. Third, refraction variation will set a fundamental limit on the accuracy of any claimed archaeoastronomical alignment. Fourth, methods for aligning the Great Pyramid of Cheops to an accuracy of 2' cannot involve near-horizon observations. Fifth, the historically important claim by A. Thom that British megalithic sites were used as accurate lunar observatories is shown to be wrong because the needed accuracy is much greater than can be obtained for long averaging intervals.