These two groups of prepositions might be harder for the native English speaker to learn because there's pretty much no analog to any structure in English, or there just seems to be no rhyme or reason for when a preposition takes an accusative or dative object, or when it might get funky and place itself behind its object for Schisse und Gekicher, and whether going post-position like that changes the specified case of the object or not!

Wait… what?

The DOGFU and the Datives post mentioned this concept, so let's not repeat it here. Read up, we'll wait.

Neither of the lists on this page (or even most on this site) should be considered exhaustive; these are just my favorites.

Genitives

außerhalb
outside of
entlang
along, as in entlang des Weges
entsprechend
correspondingly (but it's Dative in the post-position form!)
halber
for the sake of
innerhalb
inside of
jenseits
beyond
laut
according to
namens
by the name of
oberhalb
above
entlang
along
trotz
in spite of, despite
unterhalb
below
während
during
zwecks
for the purpose of

See how many of those are translatable into English with "of"? The Genitive case is to express possession, something we often do in English in one of two ways:

  1. with a possessive "'s"
  2. using "of" in a prepositional phrase

Check out trotz up there. Two direct translations into English come to mind: in spite "of", and "despite." des is the German Genitiv definite article! Coincidence?

Switcheroos

These guys are situational. Sometimes you can sort of explain away a tendency to take a dative or an accusative object if there is goal-oriented motion expressed in the clause (that often calls for an accusative object), but other times, it's apparently just determined by convention. Like when you're talking about something: über die Grammatik reden. What good reason is there for über to make Grammatik its pseudo-direct object, as opposed to its pseudo-indirect object? Let us know in the comments when you find it, please.

an
on (date), to (when addressing something to someone), at (a fixed position, like am Hauptbahnhof)
auf
to, onto, on top of
in
in, into
hinter
behind
unter
under, underneath
vor
in front of, before
zwischen
between, in between

These are tough. You get these right, and no native speaker is likely to pat you on the back and go "oh yeah, man, you're ROCKIN' out on those prepositions!". But they sure will notice if you get them wrong. Sadly, the only rules of thumb I can offer are:

  1. When the prepositional phrase is describing motion and a change of location, go Accusative. When the location of the action in the prepositional phrase is unchanging, go Dative. Beispiele:
    Ich gehe ins Zimmer. I'm walking into the room.
    Ich gehe im Zimmer. I'm walking inside the room.
  2. When in doubt in figurative expressions, like über die Grammatik reden, where there is no motion to speak of, go Accusative.

A native speaker friend-of-a-friend once told me in flawless Standard American English

You know you're really good at a foreign language when the native speakers stop complimenting you on it.1

This is one of those things that you can't expect praise for, and instead at best can only hope for no (obvious) screw-ups. Like working in IT.

1. And then he remarked on how good my German was. Hmmph. ↑