In some ways, Bill Gates, Microsoft’s founder and first C.E.O., and Satya Nadella, a longtime employee who last February became the company’s third C.E.O., are alike. They are both engineers, and they are both businessmen. In the language of the Myers-Briggs personality test, designed to measure how people perceive the world and make decisions, they both strike others as NTs, meaning they are highly rational seekers of knowledge. So perhaps they’ll work together seamlessly to run Microsoft, helping it meet the huge challenges it now faces. Gates devoted most of his time and energy during the last six years to his $40 billion philanthropic effort, the Bill & Melinda Gates Foundation, but when Microsoft announced Nadella’ s appointment on February 4, he decided to devote 30 percent of his time to Microsoft.

There’s a sense in the world outside Redmond, Washington, that Microsoft’s best days are behind it, that the sprawling colossus, which employs more than 100,000 people, doesn’t know what it is, or even what it wants to be. Gates and Nadella are adamant that’s not the case, and they are both adept at the sort of big-picture corporate-speak designed to persuade people that the company not only has its act together but also has a vision. In their view, this new world of unlimited computing power, where your devices can connect you anytime, anywhere, should rightfully belong to Microsoft. They even have a catchphrase: “Re-inventing productivity.”

When I ask them what excites them most, Nadella looks at Gates. “You want me to start?” “Sure,” Gates replies. This is, says Nadella, “a great world. It’s not lost on a few other people who are capable of exploiting that world. But the thing . . . what is scarce in all of this abundance is human attention. And whoever does the best job of building the right software experiences to give both organizations and individuals time back so that they can get more out of their time, that’s the core of this company—that’s the soul. That’s what Bill started this company with. That’s the Office franchise. That’s the Windows franchise. We have to re-invent them. . . . That’s where this notion of re-inventing productivity comes from.”

“Is software the most exciting industry in the world?,” Gates says, taking up the thread. “Absolutely. You know, when it comes to vision, speech, handwriting, screens that are going to be pervasive, that are going to let you navigate information in rich new ways, in ways that you understand your customer, what’s going on with your products. . . . We’re not even a third of the way towards empowering workers even to the dream that goes back to the start of the company.” He adds, “The opportunity is pretty incredible. And the original idea of having great software people and broad software products and Office being the primary tool that people look to across all these devices, that’ s as true today and as strong as ever.”

“The way I think about success is our relevance,” says Nadella.

Relevance, however, is exactly what Microsoft doesn’t have, according to its critics. “The Irrelevance of Microsoft” is actually the title of a blog post by an analyst named Benedict Evans, who works at the Silicon Valley venture-capital firm Andreessen Horowitz. On his blog, Evans pointed out that Microsoft’s share of all computing devices that we use to connect to the Internet, including P.C.’s, phones, and tablets, has plunged from 90 percent in 2009 to just around 20 percent today. This staggering drop occurred not because Microsoft lost ground in personal computers, on which its software still dominates, but rather because it has failed to adapt its products to smartphones, where all the growth is, and tablets. Even Microsoft’s new chairman of the board, a former IBM executive named John Thompson, told Fortune last winter that “there are some attributes to Microsoft today that do look vaguely like IBM circa 1990.” That is a particularly wounding comparison, because, as any tech person knows, IBM is the company that two decades ago an aggressive young Microsoft helped topple from the pinnacle of great technology companies.