Here's a list of the modal verbs in English:

can	could	may	might	will
would	must	shall	should	ought to

Modals are different from normal verbs:

1: They don't use an 's' for the third person singular.
2: They make questions by inversion ('she can go' becomes 'can she go?').
3: They are followed directly by the infinitive of another verb (without 'to').

1. Probability:

First, they can be used when we want to say how sure we are that something happened / is happening / will happen. We often call these 'modals of deduction' or 'speculation' or 'certainty' or 'probability'.

For example:

It's snowing, so it must be very cold outside.
I don't know where John is. He could have missed the train.
This bill can't be right. £200 for two cups of coffee!

Modal Verbs of Probability

We can use these modal verbs (also called modals of deduction, speculation or certainty) when we want to make a guess about something. We choose the verb depending on how sure we are.

1: Talking about the present:

must / might / could / may / can't + infinitive

For example:

I am waiting for Julie with another friend, David.
I ask: 'Where is Julie?'
David guesses:

She must be on the bus. (I'm fairly sure this is a good guess)
She might come soon. (maybe)
She could be lost. (maybe)
She may be in the wrong room. (maybe)
She can't be at home. (I'm fairly sure this isn't true)

Notice that the opposite of 'must' is 'can't in this case.

Will / won't

We use will and won't when we are very sure:

She'll be at work now.

Should / shouldn't

Should and shouldn't are used to make an assumption about what is probably true, if everything is as we expect:

They should be there by now.
It shouldn't take long to drive here.

This use of should isn't usually used for negative events. Instead, it's a better idea to use will:

The underground will be very busy now (not: 'should be').

Can

Can is used for something that is generally possible, something we know sometimes happens:

Prices can be high in London.

Can is not used to talk about specific possibilities:

He could be on the bus (not: 'can be').

2: Using modal verbs to talk about the past:

must / might / could / may / can't + have + past participle

must have + past participle
might / might not have + past participle
could / couldn't have + past participle
may / may not have + past participle
can't have + past participle

For example:

You: Where was Julie last night?
David:

She must have forgotten about our date.
She might have worked late.
She could have taken the wrong bus.
She may have felt ill.
She can't have stayed at home.

Will / won't + have + past participle

Will and won't / will not + have + past participle are used for past certainty (compare with present use of 'will' above):

The parcel will have arrived before now.

Should + have + past participle

Should + have + past participle can be used to make an assumption about something that has probably happened, if everything is as we expect (compare with present use of 'should' above):

The train should have left by now

Could

We can use could + infinitive to talk about a general possibility in the past (compare with the use of 'can' above):

Prices could be high in the sixteenth century.

This is not used to talk about specific possibilites in the past (instead we use could + have + past participle):

He could have been working late (not: 'could be'. As this is a specific possiblity, 'could be' is present tense)

2. Ability

We use 'can' and 'could' to talk about a skill or ability.

For example:

She can speak six languages.
My grandfather could play golf very well.
I can't drive.

Modal Verbs of Ability

When we talk about ability, we mean two things.

First, we mean general ability. This is something that once you have learned you can do any time you want, like being able to read or swim or speak a language, for example.

The other kind of ability is specific ability. This mean something that you can or can't do in one particular situation. For example, being able to lift something heavy, or find somewhere you are looking for.

Present:

can / can't (for both general and specific ability)

I can play the piano.
She can speak English.
He can't drive – he's too tired.
We can't come now.

Past:

could / couldn't (for general ability)

I could read when I was four.
She could speak French when she was a child, but now she has forgotten it.
He couldn't dance at all until he took lessons.
My grandfather couldn't swim.

was able to / couldn't (for specific ability)

When the computer crashed yesterday, I was able to fix it.(not 'I could fix it')
She was able to pass the exam, even though she hadn't studied much.(not 'she could pass')
He called us because he couldn't find the house.
I couldn't open the window.

could + have + past participle (an ability someone had in the past, but didn't use)

I could have played the piano well but I didn't practise enough.
We could have come earlier.
She could have studied law, but she preferred to become a secretary.

Future:

will / won't be able to (general ability)

At the end of the course, you will be able to make your own website.
He won't be able to speak Japanese in a week! It will take months.

can / can't (specific ability)

I can help you tomorrow
I can't come to the party

3. Obligation and Advice

We can use verbs such as 'must' or 'should' to say when something is necessary or unnecessary, or to give advice.

For example:

Children must do their homework.
We have to wear a uniform at work.
You should stop smoking.

Modal Verbs of Obligation

We can use have to + infinitive, must + infinitive and should + infinitive to express obligation (something you have to do).

Present	Positive	Negative
have to / don't have to	strong obligation (possibly from outside) Children have to go to school. (sometimes 'have got to')	no obligation I don't have to work on Sundays. You don't have to eat anything you don't like.
must / mustn't	strong obligation (possibly based on the speaker's opinion) I must study today.	negative obligation You mustn't smoke here.
should / shouldn't	mild obligation or advice You should save some money.	mild negative obligation or advice You shouldn't smoke so much.

Be careful about the difference between mustn't and don't have to!
Mustn't means it's not allowed, or it's a bad idea:

You mustn't eat so much chocolate, you'll be sick

Don't have to means you don't need to do something, but it's fine if you want to do it:

I don't have to get up early at the weekend(of course, if I want to get up early, that's fine, but I can stay in bed if I want).

Past	Positive	Negative
had to / didn't have to	obligation in the past I had to wear a school uniform when I was a child.	no obligation in the past We didn't have to go to school on Saturdays.
must*	changes to 'had to'	-
should have + pp / shouldn't have + pp	a past action which didn't happen: the advice / regret is too late You should have gone to bed earlier, now you have missed the train.	a past action which didn't happen: the advice / regret is too late You shouldn't have taken that job., it was a bad idea.

* Remember 'must have done' is a modal verb of deduction or speculation, not obligation in the past. For example: Julie must have left. Her coat's not here. See modals of probabilty for more information.

4. Permission

We can use verbs such as 'can', 'could' and 'may' to ask for and give permission. We also use modal verbs to say something is not allowed.

For example:

Could I leave early today, please?
You may not use the car tonight.
Can we swim in the lake?

5. Habits

We can use 'will' and 'would' to talk about habits or things we usually do, or did in the past.

For example:

When I lived in Italy, we would often eat in the restaurant next to my flat.
John will always be late!

6. Past modals

The past modals 'could have + past participle', 'should have + past participle' and 'would have + past participle' can be confusing. I explain about them here.

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Modal Verbs