When do I use the accusative, genitive, and dative case with certain prepositions?

The Accusative: The accusative case is used for the direct object and after certain prepositions. The most common prepositions taking the accusative case can be remembered using the mnemonic acronym FUDGE BOW which stands for für, um, durch, gegen, entlang, bis, ohne and wider. Any noun following these prepositions will be in the accusative case. The Dative: The dative case is used for the indirect object of a sentence. However, it is also used with impersonal nouns and after some prepositions. The most common preposition taking the dative case can be remembered in the rhyme, “Dative prep’s are but a few, aus, bei, mit, nach, seit, von, zu, gegenüber, außer too”. Dual-case prepositions: There are some prepositions which can either take the dative or the accusative depending on the context of the sentence. These include an, auf, hinter, in, neben, über, unter, vor and zwischen. If the sentence involves motion (or more specifically directional action), i.e. if the verb suggests the motion of an object, then the preposition takes the accusative case. Ich bin in DIE Schule gegangen. (the movement of myself to the school makes the case accusative) Ich bin in DER Schule. (I am not moving to or from the school, therefore we use the dative case) The dative case is used when there is no motion of the object following the preposition. It is also used with sentences about arriving, appearing, or disappearing. This is because these verbs do not indicate directional movement. Ich kam AM (AN DEM) Bahnhof an. The Genitive: The genitive case is used to indicate possession or the English translation ‘of’. The most common prepositions taking the genitive case are (an)statt, trotz, während and wegen. Statt eines Autos hat sie ein Fahrrad gekauft. (Instead OF a car, she bought a bike) Wegen des Wetters, gingen wir ins Kino. (Because OF the weather, we went to the cinema)