The German genitive case is very special and probably the most difficult of all the German noun cases to comprehend (for native English speakers at least).
What is the Genitive?
The most important point to understand about the genitive is that is used to show possession. Whenever you ask question with whose or Who does ... belong to the genitive case might be at play.
However, notice that it might be at play, but isn't necessarily at play. This is another aspect that makes the genitive a little more difficult. Modern German speakers don't use the genitive that often and often just use the dative instead. This used to be considered not very elegant, but it is so common these days that it would be wrong to say that this isn't "correct German" (as my high school German teacher used to say).
How to use the Genitive in English
How is possession expressed in English? Well, there are essentially two ways. First, you can add an 's and say "This is my girlfriend's dog". A second very common way is to say "This is the dog of my girlfriend". In both examples, you indicate possession, but there is no hint at something like a genitive in English.
With this in mind, let's now have a look at how the genitive works in German.
Use of the Genitive
Let's have a look at the German version of the English example sentence just discussed:
This sentence indicates possession: My girlfriend owns that dog.
But the genitive case is not just about literally possessing or owning something. It is also used when I want to express relations, for example, if I wanted to say that a person is my mum's dad, I'd say:
Lastly, there are special words & prepositions that require a genitive. Very common examples that require the genitive are trotz (despite), wegen (because of), während (during), and aufgrund (due to) & anstatt (instead of).