"Puis-je savoir qui vous êtes ?"
Translation:May I know who you are?
are we going to get into the may i can i dispute. Where I live people use can more than may. Thank you for the correction, schoolmarm
It would be used more in formal settings, perhaps in offices or asked by a receptionist.
If you are trying to strike up a conversation with an attractive person at a bar you may have more luck with "bonjour, je m'appelle Martin, comment vous appelez-vous ?" and a winning smile.