The modal auxiliary verbs are:

They are used with great frequency and with a wide range of meanings. They express ideas such as willingness and ability, permission and refusal, obligation and prohibition, suggestion, necessity, promise, and intention. All modal auxiliary verbs can express degrees of certainty, probability, or possibility.

• There is no -s in the third person: He can swim. She must go.
• There is no do/does in the question: May I ask a question? Shall we go?
• There is no doesn't/didn't in the negative: You shouldn't tell lies. You won't believe this.
• They are followed by an infinitive without to. The exception is ought to.
• They don't really have past forms, infinitives, or -ing forms. Other phrases are used instead: I'd love to be able to ski. I hate having to get up early.
• They can be used with perfect infinitives to refer to the past: You should have told me that you can't swim.

• Will and won't are used to predict a future action. The truth or certainty of what is asserted is more or less taken for granted: I'll see you later. His latest book will be out next month.
• Will and won't are also used to express what we believe or guess to be true about the present — an assumption based on our knowledge of people and things: 'You've just got a text.' 'That'll be the taxi.'

• Must is used to say what we think is the most logical or rational interpretation of a situation. We don't have all the facts, so it is less certain than will: He isn't that young! He must be 60 at least!
• The negative of this use is can't: She can't have a ten-year-old daughter! She's only 21 herself.

• Should expresses what may reasonably be expected to happen — the idea of 'if everything goes according to plan': Our guests should be here soon (if they haven't got lost).
• Should in this use has the idea that we want the action to happen. It is not used to express negative or unpleasant ideas: You should pass the exam. You've worked hard.

• May expresses the possibility that an event will happen or is happening: We may go to Greece this year. We haven't decided yet.
• Might is more tentative and slightly less certain than may: It might rain. Take your umbrella.
• Learners often express these concepts of future possibility with perhaps or maybe — will and going to avoid using may and might, but they are widely used by native speakers.

• Could has a similar meaning to might: You could be right. I'm not sure. That film could be worth seeing.
• Couldn't is not used to express a future possibility. The negative of could in this use is might not.
• Couldn't has a similar meaning to can't, only slightly weaker: She couldn't have a ten-year-old daughter! She's only 21 herself.

• Can expresses ability. The past is expressed by could: I can speak three languages. I could swim when I was three.
• Other forms use be able to: I've never been able to understand her. I'd love to be able to drive.
• To express a fulfilled ability on one particular occasion in the past, could is not used. We use was able to or managed to: She was able to survive by clinging onto the wrecked boat.

• Should and ought to express mild obligation or advice. Should is much more common: You should go to bed. You look very tired.
• We used had better to give strong advice, or to tell people what to do — there can be an element of threat: You'd better get a haircut before the interview.

• Will expresses willingness. Won't expresses a refusal by either people or things: I'll help you. The car won't start.
• The past is expressed by wouldn't: My boss said she wouldn't give me a promotion.

Several modal verbs express a request.

• Must expresses strong obligation. Other verb forms use have to: I had to work hard to pass my exams.
• Must is usually associated with a more formal, written style.
• Informally, must expresses the opinion of the speaker: I must get my hair cut.
• Have to expresses an obligation based on a law or rule, or the authority of another person.
• Mustn't expresses negative obligation. Don't have to expresses the absence of obligation.
• Have got to is common in British English. It is more informal than have to.
• Related verb forms that express obligation: Visitors are required to have a visa. You aren't supposed to park on double yellow lines.

• May, can, and could are used to ask for permission: May I ask you a question? Can/Could I borrow your car tonight?
• May can be used to give permission, but it sounds very formal. Can and can't are more common.
• To talk about permission generally, or permission in the past, we use can, could, or be allowed to: Children can/are allowed to do what they want these days.