May has two forms - may which shows that the action refers to the present, sometimes to the future and might which is used when the action refers to the past.
Note: The form might can be also used in the subjunctive mood.
I might give you this money, if you give it back in a week.
Я мог бы дать тебе эти деньги, если ты мне отдашь их через неделю.
When the action refers to the future, may can be replaced by the modal expressions to be allowed or to be able according to the type of possibility.
May express possibility due to somebody's permission.
May I use your pen?
Можно я возьму вашу ручку?In this case may is replaced by the modal expression to be allowed.
You'll be allowed to go for a walk after you have done all your lessons.
Ты сможешь пойти гулять после того, как сделаешь все уроки.May can also express possibility due to circumstances but can is more common here.
The weather is fine today and we may (can) go to the country.
Погода сегодня хорошая и мы можем поехать загород.The form might can also express reproach.
You might help me to carry this heavy bag.
Ты должен помочь мне нести эту тяжелую сумку.You might have told me about it yesterday.
Ты должен был сказать мне об этом вчера.The Perfect Infinitive is used here when the action refers to the past.
May can also express somebody's supposition implying doubt.
You may be right. = You might be right.
Ты, вероятно, прав.He may (might) be still sleeping.
Вероятно, он еще спит.He may have left.
Вероятно, он ушел.They may have been waiting for us from the very morning.
Вероятно, что они ждут нас с самого утра.The forms may and might are interchangeable here because the time of the action is expressed not by the modal verb but by the form of the Infinitive.
This model may be called Secondary modality.