On April 26 and 27 and May 10--12, 1999, unusually low solar wind densities produced unusually low Alfven Mach numbers that moved the Earth's bow shock far out past its normal location. The shocks observed by the Wind spacecraft corresponded to shock subsolar distances of 45 and 42 RE, respectively, on these days which are the most distant locations at which the shock has ever been seen. Shock observations by three other spacecraft on these days along with 34 previously reported distant shocks are used to compare with the predictions of different models. A recent MHD bow shock model of Cairns and Lyon [1995> predicts the observed locations quite well as does a modified gasdynamic model of Farris and Russell [1994> if a new Mach-number-dependent shape parameter is used. A third model of Verigin et al. [1997> also predicts a shock shape and is also quite good. Bow shock predictions are limited by uncertainties in measurements of the very low densities and uncertainties in the position and shape of the magnetopause. Asymmetries in the shock shape caused by the interplanetary magnetic field direction and not accounted for by models are another likely source of uncertainty. These uncertainties make it impossible to clearly favor one theoretical model over another. ¿ 2001 American Geophysical Union