Both must and have to can be used to express the conclusion that something is certain. Note that have to is more common in American English.

• He must be mad to do this. (OR He has to be mad to do this.)
• You must be joking. (You have got to be joking.)

Conclusions about the past are usually expressed with must followed by the perfect infinitive (have + past participle).

• I hear you have been to Australia. That must have been interesting.

Necessity

Must and have to can both be used to talk about necessity. In American English, have to is more common.

• We must/have to build up a strong army to defend the country.

Obligation

Both must and have to can be used to talk about obligation. In American English have to is the normal form. British English often makes a distinction between them. Must is used mostly to talk about the feelings and wishes of the speaker and hearer. Have (got) to is used mostly to talk about obligations that come from ‘outside’ – for example from laws, regulations, agreements and other people’s orders.

Compare:

• I must stop smoking. (= I want to stop smoking.)
• I have to stop smoking. (= Doctor has asked me to stop smoking.)
• You really must go to church next Sunday. (I am telling you to.)
• Catholics have to go to church on Sundays. (Their religion tells them to.)

Will have to is used to talk about future obligation, but have (got) to is preferred when arrangements for the future have already been made.

• When you leave school you will have to find a job. (Future obligation)
• I have got to go for a job interview tomorrow. (It is already arranged.)

Had to is used to talk about past obligation. Must is used with the perfect infinitive (have + past participle) to express certainty about the past.

• Mary isn’t in her office. She had to go home. (= It was necessary for her to go home.)
• Mary isn’t in her office. She must have gone home. (= It seems certain that she has gone home.)

Must not and do not have to/ have not got to have quite different meanings. Must not is used to tell people not to do things; do not have to/ have not got to is used to say that there is no obligation.

• You must not tell him. (= Don’t tell him.)
• You don’t have to tell him. (= You can if you like but it is not necessary.)