Forming the Genitive Case of Nouns

Genitive Singular
All masculine animate nouns and neuter nouns (except for those ending in -um, like muzeum) take the –a ending in the genitive singular. Masculine inanimate nouns take either an –a or –u ending. Unfortunately, there are no clear rules indicating when each of these endings should be used. You have to memorize which nouns take which ending!
All feminine nouns and masculine nouns ending in –a take the –y ending when the stem ends in a hard or hardened consonant. When the stem ends in a soft consonant, -k, and –g, the noun takes the –i ending in the genitive singular.

Genitive Plural
All masculine nouns with the stem ending in a hard consonant take the –ów ending in the genitive plural. When the stem ends in a soft consonant, the nouns take the –i ending. When the stem ends in a hardened consonant, the nouns take the –y ending in the genitive plural. However, there are many exceptions to the above rule.
Neuter nouns and feminine nouns ending in -i or -a have no ending in the genitive plural. Exceptions to this rule are feminine nouns ending in -nia which take the - i ending, and words common to many indoeuropean languages (e.g. historia) which take a double -i.
Feminine nouns ending in a soft consonant take the -i ending, and those ending in a hardened consonant take the -y ending in the genitive plural.
Also, several alternations occur when we form the genitive case.

See and compare the following examples:

Syntactic functions of the genitive case