English has one word for two jobs: "sorry" when you bump into someone, and "sorry" when you hear sad news. German splits them. Entschuldigung is for things you did. Tut mir leid is for feeling sorry, about your mistake or about someone else's bad luck.
Get that difference right and most apologies take care of themselves. Here are the phrases for small slips, real mistakes, formal letters, and for answering when someone apologises to you.
Everyday Apologies
Sorry.
Entschuldigung.
Excuse me, formal
Entschuldigen Sie bitte.
Excuse me, to a friend
Entschuldige bitte.
I'm sorry.
Es tut mir leid.
I'm really sorry.
Es tut mir wirklich leid.
Pardon me.
Verzeihung.
My mistake.
Mein Fehler.
Excuse me, can I get past?
Entschuldigung, darf ich mal vorbei?
Practice these words
Entschuldigung.
Entschuldigung does three jobs: it apologises for a small slip, gets someone's attention, and clears a path. Entschuldigung, ist hier noch frei? is as normal as Oh, Entschuldigung, war das Ihr Fuß?
Sorry is used in German too, especially by younger people: Oh, sorry! It sounds casual, fine with friends and in a crowded tram, less so in a formal letter.
Saying What You're Sorry For
Sorry I'm late.
Entschuldigung, dass ich zu spät komme.
Sorry, I forgot.
Sorry, das habe ich vergessen.
That was my mistake.
Das war mein Fehler.
I didn't mean it like that.
Ich habe das nicht so gemeint.
I owe you an apology.
Ich muss mich bei dir entschuldigen.
I apologise for the misunderstanding.
Ich entschuldige mich für das Missverständnis.
It won't happen again.
Das kommt nicht wieder vor.
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Entschuldigung, dass ich zu spät komme.
The verb: sich bei jemandem für etwas entschuldigen, to apologise to someone for something. Ich habe mich bei ihr für die Verspätung entschuldigt.
Good to know: some Germans say you can't excuse yourself, only ask to be excused, and prefer Ich bitte um Entschuldigung. In practice Ich entschuldige mich is used everywhere and hardly anyone minds.
When Someone Apologises to You
No problem.
Kein Problem.
It's fine.
Schon gut.
Never mind.
Macht nichts.
It's not a big deal.
Ist nicht schlimm.
All good.
Alles gut.
Don't worry about it.
Mach dir keine Gedanken.
Apology accepted.
Entschuldigung angenommen.
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Kein Problem.
Alles gut is everywhere: as a reply to an apology, as "I'm fine" and as a general calming phrase. Entschuldigung angenommen sounds more formal, sometimes a little cool, and fits better after a real mistake than after a bump in the queue.
Apologies in Emails and Letters
Please excuse my late reply.
Entschuldigen Sie bitte meine späte Antwort.
Please excuse the delay.
Bitte entschuldigen Sie die Verzögerung.
We apologise for the inconvenience.
Wir bitten, die Unannehmlichkeiten zu entschuldigen.
Unfortunately, I have to cancel the appointment.
Leider muss ich den Termin absagen.
Please excuse the misunderstanding.
Entschuldigen Sie das Missverständnis.
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Entschuldigen Sie bitte meine späte Antwort.
Leider, "unfortunately", is the German way to soften bad news. Leider muss ich absagen already contains the apology, and most Germans won't add another. On building sites and closed railway lines you'll see Wir bitten, die Unannehmlichkeiten zu entschuldigen, or the even more common Wir bitten um Ihr Verständnis, "we ask for your understanding".
When It's Sympathy, Not an Apology
I'm sorry to hear that.
Das tut mir leid.
What a shame!
Wie schade!
What bad luck!
So ein Pech!
My condolences.
Mein herzliches Beileid.
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Das tut mir leid.
Watch out: don't use Entschuldigung when someone tells you about an illness or a loss. That would mean you caused it. Das tut mir leid is right. For a death, Mein herzliches Beileid is the standard phrase, spoken or on a card.
How Germans Apologise
Short and specific. A German apology names the problem and, if possible, the fix: Sorry, das habe ich vergessen. Ich schicke es dir heute noch. A long emotional apology for something small can sound odd.
Not for things that aren't your fault. English speakers often say sorry when someone else bumps into them. Germans usually don't. If you do, people won't mind, but they may look surprised.
Late is late. Being five minutes late to an appointment is worth an Entschuldigung, dass ich zu spät komme as you arrive, and a message beforehand if you can see it coming.