a. mögen can also be translated as “to like” instead of “to like to”: Das Kind mag Eis (The child likes ice cream).
In addition mag can mean “may” suggesting possibility: Das mag wahr sein. (That may be true.)
b. können can also be translated as “to know” in the sense of skills, e.g. Sie kann Deutsch. (She knows German). See also note f) below, which explains how this works.
c. sollen is often used to distance the speaker from someone else’s claim, like English "is said to," "is supposedly," or "allegedly":
Dieses Buch soll interessant sein.
This book is said to be interesting. [or:] This book is supposedly interesting.
And as demonstrated further in section g) below, you will often encounter present-tense usages of sollen with this special meaning which refer to a past event:
Sie soll auch einen dritten Brief geschrieben haben.
She allegedly wrote a third letter, too.
d. wollen has two other common usages. As you will be able to tell from context, it can mean "to claim to" rather than "to want to":
Der Professor will diese Tatsache entdeckt haben.
The professor claims to have discovered this fact.
wollen can also be used like a regular, non-modal verb (even taking a direct object), like English "to want a thing":
Er will das Buch.
He wants the book.
e. dürfen and müssen
Because müssen and its forms so closely resemble English “must,” it is easy to mistranslate it, above all in the negative sense. For example:
Wir müssen nicht nach Hause gehen.
We do not have to go home.
müssen means “have to” and the nicht negates it. Thus müssen plus a negative means “to not have to,” NOT “must not.” Thus it would be a common mistake by English speakers to misunderstand this example as “We must not go home.”
The same danger applies when translating dürfen:
Wir dürfen nicht nach Hause gehen.
We are not allowed to go home. [or:] We must not go home.
An English speaker might misunderstand this sentence as: “We are allowed to not go home.”
f. Implied infinitives: More often than in English, in German you may see modal verbs used in a sentence without any corresponding infinitive verb. In these cases, the context provides enough information to make the sentence comprehensible. Note b) above mentioned one common example (in which the implied infinitive was sprechen) and here are three more:
Willst du jetzt nach Hause?
Do you want to go home now?
Es ist kalt hier auf dem Balkon. Wir müssen bald ins Zimmer.
It’s cold here on the balcony. We’ll have to move inside soon.
Kommst du mit ins Kino? Nein, ich mag nicht.
Are you coming with (us/me) to the movies? No, I don’t want to.
g. With perfect and passive infinitives:
Modal verbs can also be used with dependent verbs that are not in infinitive form, such as to refer to a past event or with a passive-voice dependent verb. Note in these examples the difference in tense between the modal verb and its dependent verb. In the following four examples, the modal verbs are all in present tense, but modal verbs can potentially be used in any tense, independent of the tense of the dependent verb. Watch for those tense differences and adapt your understanding of the meaning accordingly. In all cases, you can count on the dependent verb appearing at the end of the clause and in infinitive form, i.e., not conjugated to match a particular subject.
With perfect infinitives:
Mein Sohn soll das Buch gelesen haben.
My son is supposed to have read the book.
Der Zug muß neulich abgefahren sein.
The train must have departed recently.
With passive infinitives:
Diese Aufgabe muß getan werden.
This exercise must be done.
Der neue Prozeß kann schnell entwickelt werden.
The new process can be developed quickly.
Note: The dependent verbs stand in final position.
h. Past participle with ge-:
It is possible to form a past participle of the modals beginning with ge– and ending in –t, i.e., gedurft, gekonnt, gemußt, gemocht, gesollt and gewollt. However, these are only used when:
i. The modal’s special meaning is used as in:
Er hat den Film gemocht.
He liked the film.
ii. The infinitive verb is not present in the sentence, only implied:
Hat er das tun können? Ja, das hat er gekonnt.
Was he able to do that? Yes, he was able to.