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Ring-like features have been observed in several debris discs. Outside themain ring, while some systems exhibit smooth surface brightness profiles (SB)that fall off roughly as r**-3.5, others display large luminosity drops at thering's outer edge and steeper radial SB profiles. We seek to understand thisdiversity of outer edge profiles under the ``natural'' collisional evolution ofthe system, without invoking external agents such as planets or gas. We use astatistical code to follow the evolution of a collisional population, rangingfrom dust grains (submitted to radiation pressure) to planetesimals andinitially confined within a belt (the 'birth ring'). The system typicallyevolves toward a "standard" steady state, with no sharp edge and SB \proptor**-3.5 outside the birth ring. Deviations from this standard profile, in theform of a sharp outer edge and a steeper fall-off, occur only when twoparameters take their extreme values: 1) When the birth ring is so massive thatit becomes radially optically thick for the smallest grains. However, therequired disc mass is here probably too high to be realistic. 2) When thedynamical excitation of the dust-producing planetesimals is so low (<e> <0.01)that the smallest grains, which otherwise dominate the total optical depth, arepreferentially depleted. This low-excitation case, although possibly notgeneric, cannot be ruled out by observations. Our "standard" profile provides asatisfactory explanation for a large group of debris discs with outer edges andSB falling as r**-3.5. Systems with sharper outer edges, barring otherconfining agents, could still be explained by ``natural'' collisional evolutionif their dynamical excitation is very low. We show that such a dynamically-coldcase provides a satisfactory fit for HR4796A

Outer edges of debris discs: how sharp is sharp?