A. The modal verbs are can, could, may, might, shall, should, will and would.
a. The modals are used to show that we believe something is certain, probable or possible, talking about ability, asking permission making requests, and so on.

A1. Possibility: We use the modals ‘could’, ‘might’ and ‘may’ to show that something is possible in the future (but not certain). (e.g. They might come later. (= Perhaps/Maybe they will come later.))
a. We use ‘could have’, ‘might have’ and ‘may have’ to show that something was possible now or at some time in the past. (e.g. It’s ten o’clock. They might have arrived now.)
b. use the modal ‘can’ to make general statements about what is possible. (e.g. It can be very cold in winter. (= It is sometimes very cold in winter))
c. use the modal ‘could’ as the past tense of ‘can’. (e.g. It could be very cold in winter. (= Sometimes it was very cold in winter.))
d. Impossibility: We use the negative ‘can’t’ or ‘cannot’ to show that something is impossible. (e.g. That can’t be true. You cannot be serious.)
e. We use ‘couldn’t/could not’ to talk about the past. (e.g. We knew it could not be true. He was obviously joking. He could not be serious.)

A2. Probability: We use the modal ‘must’ to show we are sure something to be true and we have reasons for our belief. (e.g. It’s getting dark. It must be quite late. You haven’t eaten all day. You must be hungry.)
a. We use ‘must have’ for the past. (e.g. They hadn’t eaten all day. They must have been hungry. You look happy. You must have heard the good news.)
b. use the modal ‘should’ to suggest that something is true or will be true in the future, and to show you have reasons for your suggestion. (e.g. Ask Miranda. She should know. It’s nearly six o’clock. They should arrive soon.)
c. use ‘should have’ to talk about the past. (e.g. It’s nearly eleven o’clock. They should have arrived by now.)

A3. Ability: We use ‘can’ to talk about someone’s skill or general abilities. (e.g. She can speak several languages.)
a. We use ‘can’ to talk about the ability to do something at a given time in the present or future. (e.g. You can make a lot of money if you are lucky. Help. I can’t breathe. They can run but they can’t hide.)
b. use ‘could’ to talk about past time. (e.g. She could speak several languages. They couldn’t dance very well.)
c. use ‘could have’ to say that someone had the ability/opportunity to do something, but did not do it. (e.g. She could have learned Swahili, but she didn’t have time. I could have danced all night (but didn’t).)

A4. Permission: We use ‘can’ to ask for permission to do something (e.g. Can I ask a question, please? Can we go home now.), and to give permission. (e.g. You can go home now if you like. You can borrow my pen if you like.)
a. ‘could’ is more formal and polite than can. (e.g. Could I ask a question please? Could we go home now?)
b. ‘may’ is another more formal and polite way of asking for permission and giving permission (e.g. May I ask a question please? May we go home now? Students may travel free.)

A5. Instructions and requests: We use ‘could you’ and ‘would you’ as polite ways of telling or asking someone to do something. (e.g. Could you take a message please? Would you carry this for me please? Could I have my bill please?)
a. can and ‘will’ are less polite. (e.g. Can you take a message please? Will you carry this for me please?)

A6. Suggestions and advice: We use ‘should’ to make suggestions and give advice. (e.g. You should send an email. We should go by train.)
a. We use ‘could’ to make suggestions. (e.g. We could meet at the weekend. You could eat out tonight.)
b. We use ‘conditionals’ to give advice. (e.g. Dan will help you if you ask him.)
c. Past tenses are more polite. (e.g. Dan would help you if you asked him.)

A7. Offers and invitations: We use ‘can I ~” and to make offers. (e.g. Can I help you? Can I do that for you?)
a. We can also use ‘shall I ~’ (e.g. Shall I help you with that? Shall I call you on your mobile?)
b. We sometime say ‘I can ~’ or ‘I could ~’ or ‘I’ll (I will) ~’ to make an offer. (e.g. 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.)
c. use ‘would you like (to) ~’ for invitations. (e.g. Would you like to come round tomorrow? Would you like another drink?)
d. use ‘you must’ or ‘we must’ for a very polite invitation. (e.g. You must come round and see us. We must meet again soon.)

A8. Obligation and necessity: We use ‘must’ to say that it is necessary to do something. (e.g. 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.)
a. We use ‘had to’ for this if we are talking about the past. (e.g. Everyone had to bring something to eat. We could wear what we liked, but we had to look neat and tidy.)