The past participle used to form the perfect aspect

The past participle is used to form the perfect aspect (have + past participle) in perfect tenses, infinitives, gerunds and the perfect participle:

He didn't leave until he had talked with the boss. (past perfect tense)
I haven't smoked today. (present perfect tense)
He may have caught the train. (modal perfect)
You needn't have been so rude. (modal perfect)
If we had booked our flight earlier, it would have been cheaper. (past perfect subjunctive, modal perfect)
Lucy was assumed to have left the day before. (perfect infinitive)
I am glad to have helped you. (perfect infinitive)
After being stopped by the police, George admitted having drunk two pints of beer. (perfect gerund)
Having taken the wrong turn, he ended up in a dangerous neighbourhood. (perfect participle)