ability, permission, requests and advice

The modal verbs are can, could, may, might, shall, should, will and would.
The modals are used to do things like talking about ability, asking permission making requests, and so on.

Ability:

We use can to talk about someone’s skill or general abilities:

She can speak several languages.
He can swim like a fish.
They can’t dance very well.

We use can to talk about the ability to do something at a given time in the present or future:

You can make a lot of money if you are lucky.
Help. I can’t breathe.
They can run but they can’t hide.

We use could to talk about past time:

She could speak several languages.
They couldn’t dance very well.

We use could have to say that someone had the ability/opportunity to do something, but did not do it:

She could have learned Swahili, but she didn’t have time.
I could have danced all night [but didn't].

Permission:

We use can to ask for permission to do something:

Can I ask a question, please?
Can we go home now.

could is more formal and polite than can:

Could I ask a question please?
Could we go home now?

may is another more formal and polite way of asking for permission:

May I ask a question please?
May we go home now?

We use can to give permission:

You can go home now if you like.
You can borrow my pen if you like.

may is a more formal and polite way of giving permission:

You may go home now, if you like.

We use can to say that someone has permission to do something:

We can go out whenever we want.
Students can travel free.

may is a more formal and polite way of saying that someone has permission:

Students may travel free.

Instructions and requests:

We use could you and would you as polite ways of telling or asking someone to do something:

Could you take a message please?
Would you carry this for me please?
Could I have my bill please?

can and will are less polite:

Can you take a message please?
Will you carry this for me please?

Suggestions and advice:

We use should to make suggestions and give advice:

You should send an email.
We should go by train.

We use could to make suggestions:

We could meet at the weekend.
You could eat out tonight.

We use conditionals to give advice:

Dan will help you if you ask him.

Past tenses are more polite:

Dan would help you if you asked him.

Offers and invitations:

We use can I… and to make offers:

Can I help you?
Can I do that for you?

We can also use shall I …

Shall I help you with that?
Shall I call you on your mobile?

We sometime say I can ... or I could ... or I’ll (I will) ... to make an offer:

I can do that for you if you like.
I can give you a lift to the station.
I’ll do that for you if you like.
I’ll give you a lift to the station.

We use would you like (to) ... for invitations:

Would you like to come round to morrow?
Would you like another drink?

We use you must or we must for a very polite invitation:

You must come round and see us.
We must meet again soon.

Obligation and necessity

We use must to say that it is necessary to do something:
You must stop at a red light.
Everyone must bring something to eat.
You can wear what you like, but you must look neat and tidy.
I’m sorry, but you mustn’t make a noise in here.
We use had to for this if we are talking about the past:
Everyone had to bring something to eat.
We could wear what we liked, but we had to look neat and tidy.