Must

Obligation

"Must" is commonly used to express obligation, similar to "have to". Many text books explain the possible difference between the two as internal (must) and external (have to) obligation. Perhaps a better way to think about it is an obligation we agree with (must) and an obligation imposed upon us (have to)

We must prepare dinner before they arrive

You must wear your seatbelt in the car

Tickets must be retained for inspection

We DO NOT use "must" to talk about obligation in the past. To talk about that we use "Had to". 

The negative form is used to talk about things that are prohibited, not permitted, not allowed. This is different to the negative of Have to (Don't have to) which expresses a lack of obligation.

Prohibition

"Must not" is a form of talking about things that are not allowed, or prohibition:

You must not drink and drive

He must not have an ice cream 

It is very common to see this structure in notices and public instructions:

Passengers must not cross the yellow line

Luggage must not be left unattended

Deduction & Certainty

We often use "must" when we have considered a situation and come to a conclusion based on the evidence we have observed, known as deduction:

He must be exhausted after all that running

They must be devastated, they worked so hard to refurbish the house and then a freak storm destroyed it.

This function can also be used to talk about deductions on past events using "must" + have + participle

He must have taken the train becuase if he had driven he would be here by now

They must have decided to change house some time ago

We must have left the presents in the car

Invitation/Encouragement

We sometimes use "must" to encourage someone to do something (sometimes considered strong advice) or make an invitation:

You must come and visit soon

You must try some of this meatloaf, it is wonderful!

You must see them play live, it is an unforgettable experience.

Criticism

When we wish to express criticism of behaviour, we can use "must" in a question form to do so:

Must you eat with your mouth open?

Must you always come home so late?