He died in Seattle from complications of non-Hodgkin's lymphoma, his company Vulcan Inc. announced.
Gates said he was heartbroken about the loss of one of his "oldest and dearest friends."
"Personal computing would not have existed without him," Gates said in a statement.
"But Paul wasn't content with starting one company. He channeled his intellect and compassion into a second act focused on improving people's lives and strengthening communities in Seattle and around the world. He was fond of saying, 'If it has the potential to do good, then we should do it,'" Gates wrote.
Microsoft CEO Satya Nadella called Allen's contributions to the company, community and industry "indispensable."
"As co-founder of Microsoft, in his own quiet and persistent way, he created magical products, experiences and institutions, and in doing so, he changed the world," Nadella wrote on Twitter.
Allen, an avid sports fan, owned the Portland Trail Blazers and Seattle Seahawks.
Over the course of several decades, Allen gave more than $2 billion to a wide range of interests, including ocean health, homelessness and advancing scientific research.
"Millions of people were touched by his generosity, his persistence in pursuit of a better world, and his drive to accomplish as much as he could with the time and resources at his disposal," Vulcan CEO Bill Hilf said in a statement.
Allen was on the list of America's wealthiest people who pledged to give away the bulk of their fortunes to charity. "Those fortunate to achieve great wealth should put it to work for the good of humanity," he said.
Allen and Gates met while attending a private school in north Seattle. The two friends would later drop out of college to pursue the future they envisioned: A world with a computer in every home.
Gates so strongly believed it that he left Harvard University in his junior year to devote himself full-time to his and Allen's startup, originally called Micro-Soft. Allen spent two years at Washington State University before dropping out as well.
They founded the company in Albuquerque, New Mexico, and their first product was a computer language for the Altair hobby-kit personal computer, giving hobbyists a basic way to program and operate the machine.
After Gates and Allen found some success selling their programming language, MS-Basic, the Seattle natives moved their business in 1979 to Bellevue, Washington, not far from its eventual home in Redmond.
Microsoft's big break came in 1980, when IBM Corp. decided to move into personal computers and asked Microsoft to provide the operating system.
Gates and Allen didn't invent the operating system. To meet IBM's needs, they spent $50,000 to buy one known as QDOS from another programmer, Tim Paterson. Eventually the product refined by Microsoft - and renamed DOS, for Disk Operating System - became the core of IBM PCs and their clones, catapulting Microsoft into its dominant position in the PC industry.
The first versions of two classic Microsoft products, Microsoft Word and the Windows operating system, were released in 1983. By 1991, Microsoft's operating systems were used by 93 percent of the world's personal computers.
The Windows operating system is now used on most of the world's desktop computers, and Word is the cornerstone of the company's prevalent Office products.
Gates and Allen became billionaires when Microsoft was thrust onto the throne of technology.
With his sister Jody Allen in 1986, Paul Allen founded Vulcan, the investment firm that oversees his business and philanthropic efforts. He founded the Allen Institute for Brain Science and the aerospace firm Stratolaunch, which has built a colossal airplane designed to launch satellites into orbit. He has also backed research into nuclear-fusion power.
Allen also funded maverick aerospace designer Burt Rutan's SpaceShipOne, which in 2004 became the first privately developed manned spacecraft to reach space.
The SpaceShipOne technology was licensed by Sir Richard Branson for Virgin Galactic, which is testing a successor design to carry tourists on brief hops into lower regions of space.
Branson tweeted Monday: "So sad to hear about the passing of Paul Allen. Among many other things he was a pioneer of commercial space travel. We shared a belief that by exploring space in new ways we can improve life on Earth."
When Allen released his 2011 memoir, "Idea Man," he allowed 60 Minutes inside his home on Lake Washington, across the water from Seattle, revealing collections that included the guitar Jimi Hendrix played at Woodstock to vintage war planes and a 300-foot yacht with its own submarine.
Allen served as Microsoft's executive vice president of research and new product development until 1983, when he resigned after being diagnosed with Hodgkin's lymphoma.
"To be 30 years old and have that kind of shock - to face your mortality - really makes you feel like you should do some of the things that you haven't done yet," Allen said in a 2000 book, "Inside Out: Microsoft in Our Own Words."
Two weeks ago, Allen announced that the non-Hodgkin's lymphoma that he was treated for in 2009 had returned and he planned to fight it aggressively.
"My brother was a remarkable individual on every level," his sister Jody Allen said in a statement. "Paul's family and friends were blessed to experience his wit, warmth, his generosity and deep concern," she added.
Allen never married or had children.
His influence is firmly imprinted on the cultural landscape of Seattle and the Pacific Northwest, from the bright metallic Museum of Pop Culture designed by architect Frank Gehry to the computer science center at the University of Washington that bears his name.
In 1988 at 35, he bought the Portland Trail Blazers professional basketball team. He told The Associated Press that "for a true fan of the game, this is a dream come true."
He also was a part owner of the Seattle Sounders FC, a major league soccer team, and bought the Seattle Seahawks. Allen could sometimes be seen at games or chatting in the locker room with players.
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