You are probably familiar with the name of the German governmental department responsible for relations with foreign countries: das Auswärtige Amt. This is the German equivalent of the Department of State in the United States or the Foreign Office in the United Kingdom. The adverb auswärts, however, is often used in sports: 

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Und wir fangen noch auswärts an...

And we are still starting with an away game...

Caption 49, Basketball - Deutsche Bank Skyliners

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A literal translation of the above would render auswärts as only "away," but for clarity it's been translated for meaning as "an away game." In the next captions, you see the word as part of some nominalizations:

In dieser Saison gab es beim zweimaligen Aufeinandertreffen für beide Seiten einen Auswärtssieg.

In this season there was, during the two-time clash, an away victory for both sides.

Captions 40-41, Basketball - Deutsche Bank Skyliners

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In der vergangenen Auswärtspartie durfte der US-Amerikaner schon 'ran.

In the last away game, the American could already participate.

Caption 7, Deutsche Bank Skyliners - Basketball-Bundesliga

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Typisch Auswärtsmannschaft.

Typical away team.

Caption 32, Großstadtrevier - St. Pauli rettet HSV - Part 9

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The adverb auswärts has some practical usages outside of sports, however! Let's take a look at some examples as shown on the German dictionary site Duden: 

Lass uns doch auswärts essen!
But let's eat out! 

Or optionally:

Lass uns doch auswärts essen gehen!
But let's go out to eat! 

This is a very good one to know when you're tired of cooking and washing dishes at home!

Viele Schulkinder kommen von auswärts.
Many schoolchildren come from elsewhere.

This usage of auswärts means von einem anderen Ort or "from another place."

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Further Learning
Go to Yabla German and take a look at some of the links above to see how auswärts is used in a real-world context.