Abstract. It is remarkable that after many thousands of years of pastoralism in arid and semi-arid savannas, there is still no clear answer to the basic question: Does livestock grazing have more than a marginal effect on the dynamics of arid and semi-arid land vegetation? A small study of semi-arid savanna vegetation along a spatial gradient in grazing pressure, repeated three times over 19 years is used as a basis for discussing the behaviour of dry land vegetation under heavy grazing. Three basic theories are compared: (1) the theory that heavy grazing causes desertification; (2) the theory that heavy grazing causes no directed change in the vegetation of arid and semi-arid rangelands; and (3) the theory that heavy grazing leads to a switch between alternative states of vegetation. On the basis of the current data the first two theories are rejected, but the conclusion is that there is much evidence for a more complex behaviour of dry rangelands under grazing than what is accommodated in any of the three theories. Probably, site-specific properties including interactive and indirect effects of herbivory in the system are important for vegetation development under grazing in arid and semi-arid rangelands.