I wish you good luck.
(you express a wish)
He wished her good night.
I wish to speak to Mr Smith.
(a polite form of 'want')
I don't wish to be rude, but ...
(I don't mean to be rude; It's not my intention to be rude ...)
She wished herself a million miles away.
I wish I was taller.
(to want something to happen even though it is unlikely or impossible)
This may or may not be true.
This book may help you with your studies.
May I help you?
May I come in? - Yes, you may come in if you wish
(polite way of asking a question or making a comment)
He needs to read more books so that he may have a better education.
(indicates the purpose of something - formal)
He may be a good father but he's a bad politician.
Student or Learner