Modal verbs are auxiliary verbs (also known as "helping verbs"). Normally modal verbs cannot work alone and must work with a main verb. The so-called "semi-modals" work partly like modals and partly like main verbs.
- can, could, will, would, shall, should, may, might
- must, ought (to)
can, could, be able to
can and could are modal auxiliary verbs. Be able to is not an auxiliary verb (it uses the verb be as a main verb). We include be able to here for convenience.
have to, must
must is a modal auxiliary verb. Have to is not an auxiliary verb (it uses the verb have as a main verb). We include have to here for convenience.
shall versus will
People sometimes say that there is no difference between shall and will, or even that today nobody uses shall (except in offers such as Shall I call a taxi?). They say the same thing about should, but it's not really true.
We use the modal auxiliary verb would mainly to talk about the past, talk about the future in the past and express the conditional mood.
We use the modal auxiliary verb should mainly to give advice or make recommendations, talk about obligation or talk about probability and expectation.