Which cases do I use with prepositions?

1) These prepositions take the dative case:

aus, außer, bei, mit, nach, seit, von, zu, gegenüber

2) These prepositions take the accusative case:

bis, durch, entlang, für, gegen, ohne, um, wider

3) These prepositions take either the dative or the accusative case depending on the context:

an, auf, hinter, in, neben, über, unter, vor, zwischen

As a general rule, use the accusative case if movement is involved and the dative case if there is no movement.

For example:

a) der Mann geht über die Brücke (The man is going over the bridge)

Movement involved = accusative case

b) die Katze ist unter dem Tisch (The cat is under the table)

No movement involved = dative case

4) Some verbs contain a preposition and take a specific case.  See this website for examples of verbs with prepositions:

https://en.wikibooks.org/wiki/German/Grammar/Prepositions_and_Postpositions