“A category of VERB that regularly accompanies full verbs such as write, run, shoot, is in is writing, has in has run, may in may be shooting. In English, auxiliary verbs are customarily divided into: (1) The primary auxiliaries be, have, do. (2) The modal auxiliaries or MODAL VERBS can, could, may, might, shall, should, will, would, must. The marginal modal auxiliaries are also called semi-modals, are dare, need, ought to, used to. They are called marginal because they do not share all the properties of the others or do not do so regularly. Auxiliaries have four properties: (1) They are used with the negative not to make a sentence negative: Frank may buy me a sweater/may not buy me a sweater. (2) They form questions by changing positions with the subject: Wendy has invited me/Has Wendy invited me? (3) To avoid repetition they can occur without a full verb: Has Jonathan written to you yet?—Yes, he has. (4) They can emphasize the positive, in which case they carry the accent: David may not be there—His mother told he WILL be there. The same properties apply to be as a full verb (Jonathan isn’t tired) and particularly in British English as an alternative to have as a full verb (I haven’t a headache). In the absence of any other auxiliary verb do is introduced for these functions: Leslie didn’t tell Doreen; Did Leslie tell Doreen?: Yes, he did; he DID tell her.
The auxiliary be is used to form, with a following –ing participle, the progressive (is employing, may have been proving) and with a following –ed participle the passive (is employed, may have been proved). The auxiliary have is used with the a following -ed participle to form the perfect (has employed, may have been proved). The modal auxiliaries convey notions such as possibility, obligation, and permission. They are the only verbs not to have a distinctive third-person form in the present: He can/They can contrasts with He is/They are, He has/They have, He sees, They see. Like auxiliary do they are always the first verb in a verb phrase (should have apologized, could be making, did tell) and are followed by the bare infinitive. In standard English, two modal auxiliaries cannot co-occur, but they can in some non-standard varieties, such as Appalachian English, They might could dance.”
Excerpted from: McArthur, Tom. The Oxford Concise Companion to the English Language. New York: Oxford University Press, 2005.