It is an offence for a person under the age of 18 to have a crossbow in their possession and/or its component parts, unless they are under the supervision of a person aged 21 or over. This applies to all places not just in public places and also applies to hiring a crossbow.
The legislation only applies to crossbows, which have a certain strength, a draw weight of over 1.4 kilograms.
However, even if you are over 18 then there are only limited circumstances when you would be able to possess it without falling foul of the law on offensive weapons. Please see Q338 for more details of offensive weapons.
In addition, the bolts used in a crossbow might also fall within the scope of an offence of pointed article/blades. Please see Q337 for more information.