(Fortune Magazine) -- Arkady Volozh, СЕО (Поисковая оптимизация SEO of Yandex, Russia's largest online-search company, is playing with a set of nesting dolls (for real!). Instead of the traditional folk decoration, though, these figurines are outfitted with the names of Internet companies doing business in Russia. The first and biggest doll has Yandex emblazoned on its belly in bold red and black letters. A smaller doll bears the Google logo, followed by one representing Rambler, Russia's other homegrown search engine. "We were charitable with these dolls," Volozh says. "If we had been honest, we would have left the second doll blank and made Google third. We're that much bigger than them."

Having conquered search in Russia, Yandex is setting its sights on the U.S. The Moscow-based company is opening Yandex Labs not far from Google (GOOG, Fortune 500) headquarters in Mountain View, Calif. Volozh says he'll staff the office with 20 or so engineers to index Web pages for a Russian audience and keep abreast of technology developments that bubble up in Silicon Valley. Volozh certainly will be going after some of the same tech superstars Google recruits, but some wonder whether Yandex has grander ambitions, including a play for some of Google's U.S. market share. One possibility: grabbing part of the growing market for image searches.

Volozh denies that he's chasing Google. In fact, Yandex, which got its start in the late 1980s, long before Google founders Sergey Brin and Larry Page had even met, arguably has the superior search technology. Yandex's search algorithm is rooted in the highly inflected and very peculiar Russian language. Words can take on some 20 different endings to indicate their relationship to one another, and "while this makes the language precise," says MIT linguistics professor David Pesetsky, "it makes search extremely difficult." Google fetches the exact word combination you enter into the search bar, leaving out the slightly different forms that mean similar things. Yandex has found a way to catch them all.

As a result, Yandex controls 56% of the search market in Russia (compared with Google's 23%), boasts an impressive two-thirds of all revenue from so-called search ads, and draws more than three billion hits a month. Last month Firefox dropped Google as its default search engine in Russia in favor of Yandex, putting Yandex on track to conquer still more of the market.

And Yandex (short for "yet another indexer") continues to innovate. While some of its services are similar to offerings available in the U.S. (blog rankings, online banking), it also has developed some applications that only Russians can enjoy, such as an image search engine that eliminates repeated images, a portrait filter that ferrets out faces in an image search, and a real-time traffic report that taps into users' roving cellphone signals to monitor how quickly people are moving through crowded roads in more than a dozen Russian cities. Says noted venture capitalist and Yandex board member Esther Dyson: "What I love about Volozh is that he's never been complacent."

Volozh has aspired to run a transparent operation from the beginning, stocking his board with reputable folks like Dyson and forgoing the favor-seeking that is common in Moscow. Indeed, the fact that the SEC gave Yandex the green light to list on Nasdaq says a lot, according Nasdaq managing director Paulina McGroarty. "It shows that the management was ready to show the world that they have that stamp of approval," she says.
Changing plans

The financial crisis derailed Yandex's IPO plans, but it has helped the company stay clear of some of the Kremlin-connected oligarchs who had wanted to buy in: Industrialist Alisher Usmanov was set to take a 10% stake in Yandex, but the meltdown of the Russian market has forced the controversial businessman to rethink his investments.

Volozh says his company is cash-flow positive, and as a result it doesn't need to raise more money right now. Yandex's growth certainly has been impressive: Revenue has doubled every year since 2000, and last year reported sales topped $300 million. According to Volozh, Yandex has turned down repeated buyout offers from Yahoo (YHOO, Fortune 500), Microsoft (MSFT, Fortune 500) and Google. (The companies declined to comment.) Analysts value the business at about $6 billion.

While Yandex may not be interested in serving American customers, Google is definitely eager to increase market share in Russia, which boasts the fastest-growing Internet population in Europe. Google struggled when it first began making Russian-language search available in 2001, largely because its technology failed to take into account the complexities of Russian grammar. Its market share for much of the decade hovered at a tepid 6%. Then, in 2006, Google hired a few dozen Russian engineers to address the language issues, and its market share jumped instantly. With its huge potential for growth, says Alla Zabrovskaya, who heads public relations for Google in Moscow, "Russia is a pivotal country for Google."

As the market leader, Yandex has momentum on its side, but Volozh believes the company's real edge in Russia is its local roots. "We can joke in Russian, which Google can't do," Volozh told Fortune during a meeting at Yandex's California offices. Two years ago, for example, Yandex hosted a televised live chat with then-President Vladimir Putin. The company insisted that the President answer users' most popular questions and forgo the Kremlin's usual scripted format. "When did you have sex for the first time?" was one of the most popular, and Putin had to respond. "I don't remember, but I certainly remember the last," he quipped. "I can pinpoint it down to the minute."

The home-team advantage should help Yandex maintain its market share in Russia, but it also puts a lot of pressure on the company. If Yandex were to stumble, that would be a disappointment to Russia, which, despite its highly educated population, has few successful homegrown tech companies to root for. And Volozh might even have to order a new set of nesting dolls.