Modal Verbs of Obligation

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We can use have to + infinitive, must + infinitive and should + infinitive to express obligation (something you have to do).

Present	Positive	Negative
have to / don't have to	strong obligation (possibly from outside) Children have to go to school. (sometimes 'have got to')	no obligation I don't have to work on Sundays. You don't have to eat anything you don't like.
must / mustn't	strong obligation (possibly based on the speaker's opinion) I must study today.	negative obligation You mustn't smoke here.
should / shouldn't	mild obligation or advice You should save some money.	mild negative obligation or advice You shouldn't smoke so much.

Be careful about the difference between mustn't and don't have to!
Mustn't means it's not allowed, or it's a bad idea:

You mustn't eat so much chocolate, you'll be sick

Don't have to means you don't need to do something, but it's fine if you want to do it:

I don't have to get up early at the weekend(of course, if I want to get up early, that's fine, but I can stay in bed if I want).

Past	Positive	Negative
had to / didn't have to	obligation in the past I had to wear a school uniform when I was a child.	no obligation in the past We didn't have to go to school on Saturdays.
must*	changes to 'had to'	-
should have + pp / shouldn't have + pp	a past action which didn't happen: the advice / regret is too late You should have gone to bed earlier, now you have missed the train.	a past action which didn't happen: the advice / regret is too late You shouldn't have taken that job., it was a bad idea.

* Remember 'must have done' is a modal verb of deduction or speculation, not obligation in the past. For example: Julie must have left. Her coat's not here. See modals of probabilty for more information.