MODALS

possibility / impossibility

I can / can't open the door. It's locked.

I will / won't be able to open the door.

request permission / permission / refusal of permission

Can I go out with my friends? Yes, you can. / No, you can't.

I was (not) allowed to go out.

ability / inability

They can / can't speak Chinese

They will / won't be able to speak Chinese after 1 lesson.

disbelief (negative)

After such a quiet day as today, you can't be tired.

Help

Can you help me?

possibility (in the past)

I could / couldn't open the door. It was locked.

I was (not) able to open the door.

request permission (polite)

Could I go out with my friends?

ability (in the past)

They could / couldn't speak Chinese when they were children.

They were (not) able to speak Chinese when they were children.

disbelief (negative)

After such a quiet day as yesterday, you couldn't be tired.

Help (polite)

Could you help me?

obligation (self, internal)

I must tell you the truth.

prohibiton (negative)

You mustn't touch the paintings in a museum

belief

After such a day, you must be tired.

obligation (external)

You have to pay your taxes.

no obligation (negative)

You don't have to use all the boxes.

passive meaning

I will / am going to / had / have my car repaired.

prediction

We should earn more money next year.

advice (weak obligation)

You should / should't work harder.

permission / request permission (very polite)

May I go out with my friends? Yes, you may go out.

offers (polite)

May I take your coat?

doubt ( = it's possible)

She may come tonight.

hypothetical possibility (in the past)

I might have opened the door, but it was locked.

doubt (it's possible but unprobable)

She might come tonight.

unnecessary and performed action = waste of time

It's raining now! I needn't have watered the garden.

unnecessary but not performed action

You didn't need to tell her since she already knew the result.

future events

I am to go on holiday tomorrow.

I was to go on holiday tomorrow.

obligation (formal)

Visitors are to leave the museum at 6pm.