If you are in the UK, then the following link summarises the law:
As you can see, even a sub-50cc moped requires licensing ...
The only powered road users who can get away without licensing are those who ride bicycles which comply with the sub-20mph (I forget the official naming) electrical assist devices and mobility scooters.
You can, however, ride any power bike you want on private ground / ground you have permission to be on etc. This does not necessarily include public areas such as parks and you are entirely liable for any damage/injury you cause and with no insurance, you ride at your own risk.