In England, shall is used to express the simple future for first person I and we, as in "Shall we meet by the river?" Will would be used in the simple future for all other persons. Using will in the first person would express determination on the part of the speaker, as in "We will finish this project by tonight, by golly!" Using shall in second and third persons would indicate some kind of promise about the subject, as in "This shall be revealed to you in good time." This usage is certainly acceptable in the U.S., although shall is used far less frequently. The distinction between the two is often obscured by the contraction 'll, which is the same for both verbs.
In the United States, we seldom use shall for anything other than polite questions (suggesting an element of permission) in the first-person:
"Shall we go now?"
"Shall I call a doctor for you?"
(In the second sentence, many writers would use should instead, although should is somewhat more tentative than shall.) In the U.S., to express the future tense, the verb will is used in all other cases.
Shall is often used in formal situations (legal or legalistic documents, minutes to meetings, etc.) to express obligation, even with third-person and second-person constructions:
The board of directors shall be responsible for payment to stockholders.
The college president shall report financial shortfalls to the executive director each semester."
Should is usually replaced, nowadays, by would. It is still used, however, to mean "ought to" as in
You really shouldn't do that.
If you think that was amazing, you should have seen it last night.
In British English and very formal American English, one is apt to hear or read should with the first-person pronouns in expressions of liking such as "I should prefer iced tea" and in tentative expressions of opinion such as