Who wiped Jesus' face as he carried the cross?

Actually, we don't know the lady's name to wiped the face of Our Blessed Lord, tradition gives her name as "Veronica" but this is suspect for the simple reason that Veronica is likely the combination of vera (Latin: true) and eikon(Greek: image) so "Veronica" means "true image" referring to the piece of cloth that has a representation of Our Blessed Lord's face on it. As the actual history is not known either in the canonical Gospels or in the Church's Martyrology, we are just guessing as to its history. Tradition sometimes identifies her as the woman with an issue of blood (Mark 5, 25-34). The veil itself is in St. Peter's Basilica and is one of the greater relics, and may only be handled by the Canon of the Basilica.