What is special about the modal verbs?

 They are special because they behave differently from other verbs in English:

English modal verbs are used together with the base form of another verb.

Examples:

 "He might come late."

 "You may leave if you wish."

 "We must finish this on time."

English modal verbs have only one form. You don't add "-ing", "-ed" or "-s" to them.

"We must go now."

"They said we could park here."

"She can help us."

To form questions use the modal verb itself, but change the order.

 "He can fix the car tomorrow."

"Can he fix the car tomorrow?"

 "We should start packing our things."

"Should we start packing our things?"

 "She will be ten years old next month."

"Will she be ten years old next month?"

To form negative sentences use the modal verb itself and add "not" or "n't" to it.

 "He can run fast enough."

"He can't run fast enough."

 "She could lift a feather."

"She could not lift a feather."

 "I thought he would come."

"I thought he wouldn't come."

How and when do we use each of the English modal verbs?

Uses of "can" (negative: cannot, can't)

To talk about what you are able to do

 "He is so strong! He can lift that car!"

 "She can't come before four o'clock."

"Can he teach?"

To talk about a general possibility

 "The weather here can get really bad."

 "These chairs can be folded."

 "Such things can happen."

To say that something is allowed

 "He can borrow my book if he needs it."

 "You can't smoke in here".

 "You can pay with a credit card."

To make a request

 (this is is an informal use, "may" is the formal version)

 "Can you help me with my homework?"

 "Can you make some tea?"

 "Can you come here, please?"

Uses of "could" (negative: could not, couldn't)

As the past form of "can"

 "He said he couldn't come so early."

 "I couldn't remember who he was."

 "They couldn't pass the border."

To make a polite request

 "Could you open the window, please?"

 "Could you turn up the heat?"

 "Could you remind him to call?"

To show possibility ("may" and "might" are stronger)

 "She could be with her parents."

 "It could take you months to find a new place."

 "He could still win, but it's not very likely."

Uses of "may" (negative: may not)

To show possibility (it is slightly stronger than "might")

 "What he said may be true."

 "It may rain."

 "You may win the race."

To request or give permission

 (this is a formal use, "can" is the informal version)

 "You may sit down."

 "May I speak?"

 "He may not use the car."

Uses of "might" (negative: might not)

As a past from of "may"

 "The weatherman said it might rain."

 "She mentioned that she might come."

 "We agreed that it might be dangerous."

To show possibility (it is slightly weaker than "may")

 "He might pass the exam, but I wouldn't count on it."

 "We might fail, but let's not think about it."

 "I might visit on Saturday."

Uses of "must" (negative: must not, mustn't)

To show that you have to do something, for example because it is very important or because it is a rule

 "You must stop the car when the traffic light turns red."

 "You must pay your taxes."

 "She must stop drinking if she wants to keep her job."

 "I must go now, otherwise I will miss my train."

"Must not" (or "mustn't") is used to show you are not allowed to do something

 "You mustn't steal."

 "He mustn't talk to his parents like that."

 "The fruit of this bush must not be eaten because it is toxic."

To show that something is very logical or very likely to be true

 "He left at noon, so he must be there already."

 "She is not stupid, so she must have known what she was doing!"

 "They must be really rich to live in such a house."

Uses of "ought to" (negative: ought not to)

To say what is the right thing to do ("should" is the more common word)

 "In her condition, she ought to quit smoking."

 "I believe you ought to apologize."

 "He was watching TV when he ought to have been studying."

Uses of "shall" (short form: 'll, negative: shall not, shan't)

Used with "I" and "we" to talk about the future (especially in formal British English)

 "I shall leave tomorrow morning."

 "I'll never forget you."

 "We shall overcome."

 "I shan't be late again."

Used with "I" and "we" to ask questions or make suggestions

 "Shall I close the door?"

 "What shall we do tonight?"

 "Let's start, shall we?"

Uses of "should" (negative: should not, shouldn't)

To say what is the right thing to do

 "You should be helping your mother."

 "If he doesn't like the job, he should tell it to his boss."

 "If you knew you were going to be so late, you should have called."

To give advice or ask for advice

 "You should try the new restaurant down the street."

 "What should I do? Should I tell him the truth?"

 "Should I try to take the exam again?"

To show that something is likely to be true or that it is expected

 "Let's return home, dinner should be ready by now."

 "We should arrive there by twelve o'clock."

 "I should get an e-mail from him soon."

Uses of "would" (short form: 'd, negative: would not, wouldn't)

As the past form of "will" in reported speech

 "I will handle it myself." --> "He said he would handle it himself."

 "I won't be late." --> "He said he wouldn't be late."

 "She'll change her mind in the end." --> "He said she'd change her mind in the end."

To talk about an imagined situation

 "What would you do if you were a millionaire?"

 "I wish he'd take a break."

 "I would have cleaned the house, but I was too tired."

To make a polite request

 "Would you mind closing the window?"

 "Would you get me the paper, please?"

 "Would someone please answer the phone?"

To invite someone, or offer something, politely

 "Would you like a drink?"

 "We are going for a walk, would you like to join us?"

 "Would you like to meet her?"

To say that you want something or want to do something

I would like = a polite way of saying, "I want."

I would hate = a polite way of saying, "I don't want."

I would rather = a polite way of saying, "I prefer."

 "I would like a cup of coffee, please."

 "I would hate to miss this opportunity."

 "We'd rather study with you."

 - "Would you like to come with us?"

 -" I'd love to, but I can't."