
heute
today
Refers to the current day, that is, the day on which one is speaking or writing.
Heute scheint die Sonne.
The sun is shining today.
today
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today
Refers to the current day, that is, the day on which one is speaking or writing.
Heute scheint die Sonne.
The sun is shining today.

today / nowadays
Refers to the present time or the present in a broader sense, in contrast to the past; nowadays.
Heute sind Computer aus dem Alltag nicht mehr wegzudenken.
Today, computers are impossible to imagine everyday life without.

overnight
A fixed expression meaning that something happens very suddenly or in a very short time.
Erfolg stellt sich selten von heute auf morgen ein.
Success rarely happens overnight.
When 'heute' stands at the beginning of the sentence (position 1), the conjugated verb follows directly in position 2, and the subject comes only after that. This switch is called inversion.
Heute habe ich viel zu tun.
Today I have a lot to do.
To name a specific part of the day, 'heute' is combined with times of day. You say 'heute Morgen' (this morning), 'heute Mittag' (today at noon), and 'heute Abend' (this evening), but 'heute Nacht' (tonight/last night).
Ich gehe heute Abend ins Kino.
I'm going to the movies this evening.
The adjective 'heutig' is derived from the adverb 'heute'. It means 'of today' or 'present-day' and is declined before a noun.
Die heutige Jugend hat andere Interessen als frühere Generationen.
Today's youth have different interests from earlier generations.
The meaning 'heutzutage' (nowadays) can often be recognized by the fact that a comparison with the past is being made. Sometimes the word 'früher' (in the past / formerly) appears in the same or a nearby sentence.
Früher schrieb man Briefe, heute schickt man E-Mails.
In the past people wrote letters; today they send emails.
now
Jetzt and heute both refer to the present, but they indicate different time spans. Jetzt means the current moment, while heute refers to the entire present day. Use jetzt for this moment and heute when the statement applies to today as a whole.

Ich kann heute nicht telefonieren, weil ich den ganzen Tag unterwegs bin.
I can't talk on the phone today because I'll be out all day.
Ich kann jetzt nicht telefonieren, weil ich gerade im Bus sitze.
I can't talk on the phone right now because I'm on the bus.