Genitive

“Of”

The primary use of the genitive case is to express the “of” idea:

Это карта Москвы. This is a map of Moscow
Это пакет молока. This is a carton of milk.

Possession

The genitive case is used to express possession. It corresponds to the  's  in English. Notice that the possessor comes after the thing possessed:

Мы катались на машине Ивана. We drove around in John's car.
Где очки мамы? Where are Mom's glasses?

Kinship, family, and hierarchical relationships

The genitive case is used to express relationships like mother, father, boss:

Это мама Ивана. This is John's mom.
Я работаю с сестрой Ларисы. I work work with Larisa's sister.
Вчера в кино я увидел босса брата. Yesterday I saw my brother's boss at the movies.

Certain prepositions

The genitive case is used after certain prepositions, including без, после, из, от, and с (in the sense of “from”):

Я пью чай без сахара. I drink tea without sugar.
После лекции я пойду домой. After the lecture I will go home.
Я иду из магазина. I'm coming from the store.
Она идёт с почты. She is coming from the post-office.
Мы идём от Тани. We are coming from Tanya's place.

Have

The genitive case is used after the preposition у in the Russian construction that is equivalent to the English “have”.

У Ивана есть машина? Does Ivan have a car?
У Тани была ручка. Tanya had a pen.
У нас будет свободное время через неделю. We will have free time in a week.

Negation of existence

The genitive case is used to indicate that a zero quantity of something:

В нашем доме нет иностранцев. There are no foreigners in our apartment building.
На письменном столе не было принтера. There was no printer on the desk.
Скоро не будет возможности ездить за границу. Soon there won't be any opportunity to travel abroad.

This negation of existence concept also applies in sentences where someone does NOT HAVE something:

У Тани нет братьев. There are no foreigners in our apartment building.
У меня вчера не было денег. Yesterday I didn't have any money.
В будущем не будет питевой воды около этого завода. In the future there won't be any drinking water near this factory.

Quantity phrases

The genitive case is used after most words and phrases that indicate quantity

2, 3, 4

After the numbers два/две, три and четыре the genitive singular of nouns is used:

В нашем доме два холодильника. Our house has two refrigerators.
На письменном столе было три принтера. There were three printers on the desk.
Завтра у них будет четыре гостя. Tomorrow they will have four guests.

If you put an adjective between the number and the noun, then the adjective goes in the genitive plural:

два больших холодильника two large refrigerators.
три новых принтера three new printers
четыре интересных гостя four interesting guests

These rules also apply for compound numbers whose spoken forms end in два/две, три and четыре:

двадцать два больших холодильника twenty-two large refrigerators.
семьдесят три новых принтера seventy-three new printers
девяносто четыре интересных гостя ninety-four interesting guests

The genitive singular rule does not apply to the numbers двенадцать, тринадцать or четырнадцать.

Numbers 5 and above

After the numbers five through twenty the genitive plural of nouns is used:

В нашем доме пять холодильников. Our house has five refrigerators.
На письменном столе было восемь принтеров. There were eight printers on the desk.
Завтра у них будет тринадцать гостей. Tomorrow they will have thirteen guests.

If you put an adjective between the number and the noun, then the adjective goes in the genitive plural:

пять больших холодильников five large refrigerators
восемь новых принтеров eight new printers
тринадцать интересных гостей thirteen interesting guests

These rules also apply to all other numbers that do not end in forms of один, два, три or четыре.

Generic quantity words and phrases

The genitive is used after these generic quantity words:

сколько how much, how many
столько so much, so many
много much, many, a lot of
мало little, few, not much
несколько several
немного a little bit of

Examples:

Сколько машин производит фабрика? How many cars does the factory produce?
Я там видел столько больших собак! I saw so many big dogs there!
У меня много денег. I have a lot of money.
У меня мало времени. I have little time ≈ I don't have much time.
Она взяла у меня несколько яиц. She borrowed a few eggs from me.
Хочу пить немного воды. I want to drink a little water.

Packages, cartons, and boxes of goods are followed by the genitive to indicate what they contain. Weights and measures are also followed by the genitive form of the goods they measure.

Я купил пачку сигарет. I bought a package of cigarettes.
Таня купила две коробки конфет. Tanya bought two boxes of candy.
Купи бутылку водки. Buy a bottle of vodka
Мне нужно три килограмма яблок. I need three kilos of apples.
Нам нужно пять литров бензина. We need five liters of gasoline.

Date on which

The genitive case is used to express the date on which something happens. No preposition is used.

Мы были в Москве пятого числа. We were in Moscow on the fifth.
Где ты будешь тридцатого? Where will you be on the thirtieth?

If you also specify the month, then the month also goes into the genitive:

Мы были в Москве пятого января. We were in Moscow on the fifth of January.
Где ты будешь тридцатого июля? Where will you be on the thirtieth of July?

If you also specify the year, then the year also goes into the genitive:

Мы были в Москве пятого января тысяча девятьсот восемьдесят девятого года. We were in Moscow on the fifth of January.
Где ты будешь тридцатого июля две тысячи двадцать первого года? Where will you be on July 30th, 2021?