We use both must and have to to express obligation or necessity, but there is sometimes a difference between them:

We tend to prefer must:

—  when we refer to ourselves (with I/we): I really must weed this garden.

—  With you to express urgency: You must phone home at once.

—  in public notices, etc.: Cyclists must dismount.

— (= Can't you stop yourself?): Must you interrupt?

—  pressing invitations or advice: You must come and see us. You must repair that fence.

We only use must (+infinitive) to talk about the present and the future. When we talk about past obligation or necessity, we use had to.

I had to work late yesterday.

Must has no infinitive, -ing form or participles. So, when necessary, we make these forms with have to.

I'll have to work late tomorrow.

He hates having to get up early.

She's had to work hard all her life.

Note that in questions and negatives with have to we use do/does in the present simple and did in the past simple.

What time do you have to start work?

We don't have to hurry. We've got plenty of

time.

Did you have to walk home last night?

Have got to

We often use have got to instead of have to to talk about obligation and necessity. Have got to is more informal and is used primarily in spoken English. Have to is used in both formal and informal English.

We normally use have to, not have got to, for things that happen repeatedly, especially when we use one-word adverbs of frequency eg always, often. Compare:

We use got mostly in the present.   To talk about the past, we normally use had to, not had got to.

I had to work late last night.