May is used to express:
1. uncertainty, probability, possibility. Example :
Bruno may be in the board room. It's possible that Bruno is in the board room.
It's possible that dogs may not be allowed in the apartment. Perhaps dogs are not allowed in the apartment.
2. when used to express permission, may is equivalent to can or could, but is more polite and formal. Example :
May I have your name please? Could you please tell me your name?
May I smoke? Yes, you may. a more formal version of: Can I smoke? Yes, go ahead.
3. desires and wishes. Example :
May the force be with you! (Star Wars) I hope the force will be with you!