Greetings
Previet! - Hi! (informal)
Zdravzvutye - Hello. (Formal)
Poka - Bye! (Informal)
Do Svidanya - Good-bye (Formal)
Kak oo vac dela? - How are you? (Formal)
Kak dela? - How are you? (Informal)
Horosho, a oo vac? - Fine, and you?
Dobree ootra - Good Morning
Dobree vecher - Good Evening/afternoon
Dobree del - Good Day

Descriptions
horosho - fine
ploha - bad
kraceevoy - beautiful; good-looking
staree - old
novee - new
seempateechnee - cute

Family
rodetele - parents
mat - mother
otyets - father
brat - brother
sestra - sister
sobaka - dog
koshka - cat
ochen - very

Numbers
odin - one
dva - two
tree - three
chetyeree - four
pyat - five
shest - six
sem - seven
vocyem - eight
devyat - nine
dyeset - ten

Giving and receiving
Poshalsta - Please, you're welcome, here you are
spaseebo - thank you
Bolshaya Spaseebo - Literally, "Big Thank you"

Miscellaneous
Sevodnya eediot snyek - Today it is snowing (Literally, "Today is going snow")

These might not be quite so common, but they are fun nonetheless.

Kak dyela?-How are you doing?

Shtow yeshow?-Anything else?

Spasiba-Thank you

Pojalsta-You're welcome/Please(offering something)

Ya ne znaiyoo-I don't know

Ya ne ponymaiyoo-I don't understand

tselooyi minyeh-Kiss me!

Ya hochoo zilonaiya piva-I want green beer

Daiyete bomby-Give me the bomb

Mwee gavarili o tebye-We talked about you

Tee vidyelish moyio vodkoo?-Have you seen my Vodka?

Kto vash papa?-Who's your daddy?

Vot eta da!-Wow!

Ya lioblio tebye-I love you

Oo menya yest tre goleeck klownov-I have three naked clowns

Here are a bunch of phrases, with the typos fixed from previous writeups:

Common Phrases:
Thank You - Spasibo
Please/You're Welcome - Pozhalusta
Hi - Privet
Hello - Zdrastvuyte
Bye - Poka*
Goodbye - Do Svedanya (lit. till we meet)
Good Morning - Dobroyo Utro
Good Day - Dobriy Den'
Good Evening - Dobriy Vecher
Good Night - Spokoyniy Nochi

Things:
Pen - Ruchka
Pencil - Korondash*
Cow - korova*
Chicken - kuritza
Car - Mashina
House - Dom

Useful/Not So Useful Phrases:
Help! My car is on fire! - Pomogite*! Moya* mashina gorit*!
Someone wants to kill me! - Kto-to hochet menya ubit'!
Do you have any vodka? - U vas est' vodka?
I am a silly American, please do not rob me. - Ya glupiy amerikanetz, pozhaluysta ne obvaruyte menya.
I hate this country! - Ya nenavizhu etu stranu!
You are all communists! - Vi vse kommunisti!
It's very cold here. - Tut ochen holodno.
Give me a sweater. - Day mne koftu.
Help! I've got dollars! - Pomogite*! U menya est' dolari!
Where is the nearest bathroom? - Gde blizhayshiy tualet?
That's really far away! - Eto ochen daleko!
How do I get there? - Kak mne tuda popast'?
Wow, vodka is cheap here! - Wow, vodka ochen deshovaya tut!
To boldy go where no man has gone before. - Idti smelo tuda kuda nekto eshe ne poshel.
Today is a good day to die. - Sevodnya horoshiy den' umeret'.
Mary had a little lamb. - U Mary bila malenkaya ovtza.
Where can I buy tickets to get out of this frozen hell? - Gde ya mogu kupit' bileti shtob vibratsa iz etovo zamerzhshovo ada?
The end. - Konetz.

* These words are often pronounced with an 'a' sound for the 'o', e.g. poka is pronounced paka, this is true in southern regions of Russia, e.g. Moscow.

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