In essence, any SEP aircraft you want to fly, you must be endorsed to fly the aircraft make and model.
Certainly not the case in the UK, as Ghengis says
In reality, nobody is going to let a new PPL fly something relatively big and complex without some conversion training.
, but you could actually go out and buy any aircraft that you had covered differences training for and legally fly it with no further training, so, having trained on a C152 you could fly most tricycle SEP with fixed pitch prop and standard instruments without further training, more complex aircraft you need differences training for things like VP props, retracts and glass cockpits, but even this is still not type specific!