To modify the verb in the main clause:

Se disculpó por haber olvidado He apologized for forgetting

¿Recuerdas haber ido a España? Do you remember going to Spain?

To modify the adjective in the main clause:

Estoy encantada de haberte visto. I'm delighted to see you.

Era necesario haberlo terminado. It was necessary to have it finished.

With the preposition después (after):

Después de haber comido, fui al parque After eating, I went to the park

Quería viajar después de haber terminado mis estudios I wanted to travel after finishing my studies

When expressing gratitude:

Gracias por habernos ayudado. Thank you for helping us.

Gracias por haber venido. Thanks for coming.

Unlike other compound tenses, object pronouns used with the perfect infinitive are attached to the end of the auxiliary.

Después de haberte visto... After seeing you...

Gracias por haberme ayudado. Thanks for helping me.