Neutron diffraction data with high real-space resolution are reported for an extensive series of cesium germanate glasses with 2, 5, 10, 15, 18, 21, 25, and 30 mol % Cs2O, and also for pure GeO2 in its vitreous, quartz, and rutile forms. The results for pure GeO2 show that neutron diffraction can clearly identify an increase in the Ge-O coordination number above the tetrahedral value of four. The results for cesium germanate glasses give strong evidence that the Ge-O coordination number rises to a maximum of 4.36 +/- 0.03 for 18 mol % Cs2O and then declines. This behavior may be associated with the germanate anomaly in the thermophysical properties. A model is developed for the composition-dependence of the Ge-O coordination number, and according to this model the rise in the coordination number involves the formation of mostly GeO5 units, rather than GeO6 units. Our results also show that the low alkali anomaly is a longer range phenomenon, and is not associated with a preference for the formation of nonbridging oxygens for very low alkali oxide content.