Let's start out by taking a look at the genitive case for nouns as follows for the definite article "the," with the nominative case followed by the dative case:
der => des
die => der
das => des
die (plural) => der
And for the indefinite article:
ein (masculine) => eines
eine => einer
ein (neuter) => eines
Remember too that if there is no definite or indefinite article, the adjective must still take the case appropriate for its gender with the preposition. And while accusative and dative prepositions aren’t (with some exceptions) usually accompanied by altered nouns, masculine and neuter singular nouns preceded by genitive prepositions get the suffix “-s” or “-es.” Generally plural nouns don't change.
The common German prepositions that require the genitive case are anstatt or statt, auβerhalb, innerhalb, trotz, während, and wegen. The preposition statt, which is not to be confused with the separable verb stattfinden, takes the genitive case. Here, the plural die Runden becomes the genitive der Runden:
... heizte Vettel statt der geplanten drei insgesamt sechs Runden über den Asphalt.
... instead of the planned three laps, Vettel sped six laps over the asphalt.
Caption 46, Formel-1-Autorennen Sebastian Vettels HomerunPlay Caption
Here, the singular feminine noun die Wertung becomes the genitive der Wertung:
... und er fährt hier außerhalb der Wertung mit.
... and he's riding here without being scored.
Caption 35, Trial-Meisterschaft in Bensheim - Part 2Play Caption
Note that if a masculine or neuter proper noun is used, such as the continent das Asien, the proper noun has "-s" added as a suffix:
Bislang gibt es außerhalb Asiens überhaupt noch keine vergleichbare Drehscheibe.
Until now, outside of Asia, there hasn't been a comparable [yuan trading] hub at all.Play Caption
In the following, the masculine nominative ein Dom becomes the genitive eines Doms:
Wir haben vier Segmente innerhalb eines Doms.
We have four segments inside a dome.
Caption 52, Bildverarbeitung Sirius Advanced CyberneticsPlay Caption
It's also often possible to use außerhalb or innerhalb as dative prepositions by pairing them with von for the dative case: außerhalb von and innerhalb von.
Here, the nominative die Perücke, with the adjective goldene Perücke, becomes the genitive goldener Perücke. It is important to note that if no definite or indefinite article is present, the adjective takes the genitive ending:
Trotz goldener Perücke hat der einst vornehme Herr wohl schon bessere Zeiten gesehen!
Despite the gold wig, the once distinguished gentlemen has indeed seen better times!
Caption 3, Architektur Karlsruher Brunnen - Part 2Play Caption
In this next caption, the neuter nominative das Training becomes the genitive des Trainings:
Und wie ist das passiert? Während des Trainings?
And how did it happen? During the training?
Caption 33, Jenny und Alena HandballPlay Caption
And in the last of our genitive case examples, the masculine nominative der Getriebeschaden becomes the genitive des Getriebeschadens:
... wegen eines Getriebeschadens zurück in die Boxengasse.
... go back into the pit lane due to transmission damage.
Caption 43, Formel-1-Autorennen Sebastian Vettels HomerunPlay Caption
You may have noticed that the personal pronouns were not listed at the top of this lesson. That's because personal pronouns aren't generally used in formal German, however the usually genitive prepositions anstatt or statt, trotz, während and wegen are sometimes used with personal pronouns informally — but with the dative case!
Thus the singular and plural personal pronouns "I," "you," "he," "she," , "it," "we," and "they" take the dative case with the above genitive prepositions:
ich => mir
du => dir
Sie (formal "you") => Ihnen
er => ihm
sie => ihr
es => ihm
ihr => euch
wir => uns
sie => ihnen
Sie (formal "you" plural) => Ihnen
Probably the most commonly-heard example is, instead of saying "because of me" as meinetwegen, you'll often hear:
Und ihr seid auch da. Etwa wegen mir?
And you are here too. Somehow because of me?
Caption 2, Otto Waalkes: Hier kommt Otto!
Remember too that meinetwegen also has a slang usage meaning "I don't have anything against it". We'll cover the informal usage of dative personal pronouns with usually genitive prepositions in another lesson!
To recap, the common German prepositions that require the genitive case of definite and indefinite articles and nouns are anstatt or statt, auβerhalb, innerhalb, trotz, während, and wegen. Go to Yabla German to look for more examples of prepositions whose nouns, pronouns, and adjectives formally take the genitive case. Also review Part II in this series about prepositions that require the dative case.