In England and Wales, it is the requirements in terms of muzzle velocity and muzzle energy. This tends to then limit the calibres that can meet the requirements. All bullets must be expanding.
.240 is the minimum legal calibre for Red, Roe, Sika and Fallow. Muzzle energy of not less than 1700ft lbs. No minimum bullet weight.
.220 is the minimum legal calibre for CWD and muntjac. Muzzle energy of not less than 1000ft lbs, and a minimum bullet weight of 50 grains
As above, for all deer species except Roe, Scotland also has minimum bullet weights of 100 grains to consider and a muzzle energy of 1750 ft lbs and a muzzle velocity of at least 2450ft/sec.
For roe deer ONLY, bullet weight of 50 grains, ME of 1000 ft lbs, and MV of 2450 ft/sec.
If I have got any of that wrong, someone please shout - copied from the DSC1 manual, but may have been amended since then?