Yes. In UK, all mopeds must be registered to be ridden on public roads. The process of registering a moped is through a process called DVLA registration. This means that you must:
* have a valid driving license
* have insurance for the moped
* pay the appropriate vehicle registration fee
* display a valid registration plate on the moped
You can find out more about the process of registering a moped on the DVLA website.