Shall and will are two of the English modal verbs. They have various uses, including the expression of propositions about the future, in what is usually referred to as the future tense of English.
The traditional prescriptive grammar rule stated that, when expressing pure futurity (without any additional meaning such as desire or command), shall was to be used when the subject was in the first person (I or we), and will in other cases. In practice this rule is commonly not adhered to by any group of English speakers, and many speakers do not differentiate between will and shall when expressing futurity, with the use of will being much more common and less formal than shall.
Shall is widely used in bureaucratic documents, especially documents written by lawyers. Due to heavy misuse, its meaning is vague and the United States government's Plain Language group advises writers not to use the word at all.
The verb shall derives from Old English sceal. Its cognates in other Germanic languages include Old Norse skal, German soll, and Dutch zal; these all represent *skol-, the o-grade of Indo-European *skel-. All of these verbs function as auxiliaries, representing either simple futurity, or necessity or obligation.
The verb will derives from Old English willan, meaning to want or wish. Cognates include Old Norse vilja, German wollen, Dutch willen, Gothic wiljan. It also has relatives in non-Germanic languages, such as Latin velle ("wish for") and voluptas ("pleasure"), and Polish woleć ("prefer"). All of these forms derive from the e-grade or o-grade of Indo-European *wel-, meaning to wish for or desire. Within English, the modal verb will is also related to the noun will and the regular lexical verb will (as in "She willed him on").