The Russian noun for “a smile” is улыбка, and the verb is улыбаться/улыбнуться.

In Russia smiling is not required to be polite. A store clerk can walk up to you, look you in the eye, and not say a thing. If you ask her for something and she gives it to you without a word, then she has been perfectly polite: you got what you wanted without a hassle. This lack of smiling creates a very bad impression on Americans who go to Russia; they come away with the feeling that everyone in the country is rude. They are wrong. It's simply a different social standard. Russians smile when they are happy, and I don't mean happy in the sense of "Oh, here comes my 78th customer of the day, what a joy, I'll smile because I'm really interested in him as a human being!" That's an American delusion. Russians smile when they are actually happy. A Russian man smiles when he hasn't seen his girlfriend in three weeks and she shows up at the door, flings herself into his arms and offers to make dinner. A Russian woman smiles when she finds three hundred dollars on the ground and no one else is around so she gets to keep it. That's what smiling is about in Russia: actual specific happiness. It's never smiling just to be polite.

The Russians label the American smile as «дежурная улыбка» “a business smile,” the kind of smile you put on your face just because that's what Americans are in the habit of doing. That kind of smiling creates an impression of insincerity and falsehood in the Russian mind and makes people wonder, "What the heck is so-and-so up to?” There's a story of an American business coach teaching the employees of a Russian company to constantly smile at each other and say, “Have a nice day.” Three weeks after the consultant left, everyone in the office thought everyone else was out to get them. The American smiles caused mutual suspicion, a sense of falsehood; it was completely the wrong thing to teach a company in a Russian environment.

It's funny: we Americans tend to think that our smiles are a spontaneous expression that springs simply from emotion. Not so. A smile is a learned behavior which is used to mean different things in different places. I remember back in the 2001 World Series there was a game where Byung-Hyun Kim made a series of bad pitches that could have lost the game. He stood in the middle of the field staring at the ground and grinning like an idiot. The American audience totally misunderstood his smile. Some thought he was being overconfident. Some thought he was thinking it just didn't matter. But Koreans may use a smile to mask negative feelings, and in this case the feeling was shame. The guy wasn't overconfident. He wasn't making light of the situation. He was feeling like a complete failure and was thinking about how his wife and all his friends and countrymen would be ashamed of him.

The lesson to draw is this: when we first go to Russia, our American intuition will tell us that the Russians are always being rude. That intuition is false. You will have to train yourself to remember that those intuitions you have are based on American expectations of body language. Russian body language is different. Use your mind to teach your emotions how to reinterpret things.

You can find an interesting article about Russian and American smiles here. The section contrasting how Russians and Americans smile at children and pets strikes me as being slightly exagerrated, but on the whole I think there is some truth there.

PS. On October 21st, 2008 Russian singing duo «Тату» (t.A.T.u. in English releases) released a new album called «Весёлые улыбки» "Happy Smiles."

2013-12-06: I just came across another neat article on Russian smiles at Russia Beyond the Headlines (mirror). You can find the original of the article at adeptis.ru (mirror).

2014-01-09: A former student of mine, Graeme Fox, pointed out this new addition (mirror) to the discussion of Russian smiles. Point #11 is quite good; I had never thought of that aspect.