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Müssen vs. sollen vs. dürfen

The German modal verbs müssen, sollen, and dürfen all relate to obligation, duty, or permission, but they are not interchangeable. Each carries a distinct meaning that specifies the nature of the requirement or allowance, from a personal necessity (müssen) to an external command (sollen) or formal permission (dürfen).

Müssen

A1
Müssen expresses a strong necessity or an unavoidable obligation. It means must or to have to. This can be due to an internal conviction, a logical conclusion, or an external force that leaves no other choice.
Ich muss jetzt gehen, sonst verpasse ich den Zug.
(I have to go now, otherwise I will miss the train.)
Wir müssen die Rechnung bis Freitag bezahlen.
(We must pay the bill by Friday.)
Er muss sehr intelligent sein, um das zu lösen.
(He must be very intelligent to solve that.)
Alle Menschen müssen atmen, um zu leben.
(All people must breathe in order to live.)
Du musst nicht alles alleine machen.
(You don't have to do everything alone.)

Sollen

A1
Sollen indicates an obligation, a duty, or a recommendation coming from an external source. It translates to should or to be supposed to. It's often a command, advice, or expectation from another person, a rule, or a moral principle.
Der Arzt sagt, ich soll im Bett bleiben.
(The doctor says I am supposed to stay in bed.)
Du sollst deine Hausaufgaben machen.
(You are supposed to do your homework.)
Wir sollen unsere Nachbarn respektieren.
(We should respect our neighbors.)
Soll ich das Fenster für dich öffnen?
(Should I open the window for you?)
Laut Gesetz soll man hier nicht parken.
(According to the law, one is not supposed to park here.)

Dürfen

A1
Dürfen expresses permission. It means may or to be allowed to. In its negative form (nicht dürfen), it signifies a strict prohibition, meaning must not.
Darf ich hier rauchen?
(May I smoke here?)
Kinder dürfen diesen Film nicht sehen.
(Children are not allowed to watch this movie.)
Sie dürfen jetzt nach Hause gehen.
(You may go home now.)
In diesem Park dürfen Hunde ohne Leine laufen.
(In this park, dogs are allowed to run without a leash.)
Man darf nicht bei Rot über die Ampel gehen.
(One must not cross the street when the light is red.)

Summary

In short: müssen is a must (necessity from within or due to circumstances), sollen is a should (an order, advice, or expectation from others), and dürfen is a may (permission). A key distinction is nicht dürfen which means must not (prohibition), whereas nicht müssen means 'don't have to' (no necessity).