Unfortunately, we don't. Neuromancer is set in the future, but the date's a big old question mark. All we can say for certain is that it takes place in the far future, or what would have been the far future in '84. Beyond that it's anybody's guess.
The world of Neuromancer hints at a lot but doesn't reveal many of its secrets. Urban sprawl has reached an all time high in this dystopian future, and mega-corporations seem to hold more power and sway than any actual government. There was a war, but it's never said between whom or why or where. All that is mentioned is that it lasted three weeks, and there was an operation called Screaming Fist where the US tried to infiltrate Russian intelligence. Animals have become rare enough that real meat is a delicacy. In some cases, like horses, they've gone extinct altogether. Again the details are minimal. The Finn mentions a pandemic but that's all we get (7.60).
What we can say for certain is that technology dominates this future world. And we mean dominates. People get tech modifications to gain superhuman skills. Simstims allow people to view the world through the eyes of another person. Entertainment is dominated by technology like virtual reality video games. Almost every job, whether criminal or on the up and up, deals in some way with technology, particularly the communal database called cyberspace. Wait, we're still talking about the future, right?
Chiba City is where Case starts his journey, so of course we had to include a Journey reference to start this thing right.
Chiba City, Japan is a port city on Tokyo Bay (for realsies, too), but Case spends most of his time in a part of town called Night City. And we'll find out soon enough that it has that moniker for obvious reasons. During the day, "the bars down Ninsei [street are] shuttered and featureless, the neon dead, the holograms inert, waiting, under the poisoned silver sky" (1.33). But once that sun goes down, watch your butt. Those neon lights and holograms explode to life, and the typical customers of the arcades and bars include pimps, Yakuza, fences, and other members of the criminal class. It's the kind of place where you can get a gun on the fly, enjoy an illegal surgery, or even bet on a knife fight.
Case calls Chiba City a "deranged experiment in social Darwinism" with "death the accepted punishment for laziness, carelessness, lack of grace, the failure to heed the demands of an intricate protocol" (1.36-37). In short, it's an urban jungle. Survival of the fittest.
The Sprawl is the nickname of the Boston-Atlanta Metropolitan Axis (3.2). It's a huge city spreading across most of the east coast of the United States, and it's covered in geodesics, or domes that change color with the time of day (4.121). According to Case, if you "program a map to display frequency of data exchange, every thousand megabytes a single pixel on a very large screen[,] Manhattan and Atlanta burn solid white" (3.3). For a console cowboy like him, this vast trafficking of information makes the Sprawl home.
The Sprawl and Chiba City are both urban centers, but there are notable differences. Chiba City's commerce is centered on a black market that's focused more on physical items like guns, computer parts, and human enhancements. Meanwhile, the Sprawl's commerce focuses more on "mall crowds" (3.33) on the surface while its underground specialty is information. Pure, uncut digital information. The stuff that occupies cyberspace, not real space.
You can see this difference in the way the criminals of each city do business. In Chiba City, the gangsters come at you with a hit man, like the guy who went after Case at the end of chapter 2. Meanwhile, the gangsters in the Sprawl have more subtle methods. Notice how the Panther Moderns do horrific amounts of violence simply by making a few phone calls and showing a false TV broadcast. Which do you think is scarier?
Case and company don't spend a lot of time in Istanbul, but we still learn some interesting facts about the place during our time there. Istanbul is a city that has blended the old world into the new. The urban overpopulation and growth exists side by side with the history of the old world. We can see this combination in the baroque style of Topkapi Palace (7.79-80), and the goods at the bazaar. Futuristic technology exists in Istanbul since they drive an automated, fully voiced Mercedes, but it's still hindered by the rules and customs of the old world. For example, Molly's modifications are considered scandalous because she is a woman (7.37). Some things never change? (We hope not.)
Zion is a colony in space built by five workers who just up and refused to come back to the overcrowded rock that is Earth. As a result, its construction seems more slapdash than Freeside's.
The current inhabitants of Zion follow the Rastafarian ideology. One of the tenets of Rasta is the return to Africa by the descendants of slaves, where they hope to establish a free state. In a way, the Zion colony of Neuromancer is the realization of that African free state, only in space, so minus the zebras.
Freeside is what you'd get if you combined Abu Dhabi and Las Vegas, shoved them into a giant cylinder, and shot it into space. The rich and powerful either live in Freeside full-time or go there to play as often as they can.
But Freeside does have a dark side to it. The Tessier-Ashpool family uses it as a base of operations just like the Mafia used Vegas back in the day. The wealthy can use Freeside as an "orbital Geneva," meaning they can have secret back accounts for their less than stellar income earnings. There is also a seedy side to the entertainment in the form of "meat puppet" brothels and hardcore drugs.
Freeside does have hang gliding through, so you take the bad with the good.
Straylight is the personal home of the Tessier-Ashpool family. The structure, like the family, is eccentric and hidden away from the outside world. According to 3Jane, Straylight is a "desperate proliferation of structures, forms flowing, interlocking, rising toward a solid core of micromechanical decay and sabotage" (14.105). In short, it's a maze that's always being added to and reworked even as it's breaking down. When Molly infiltrates the facility, she finds parts of it are littered with antique books, sculptures, furniture, and the like, all in various states of disarray and blanketed with dust.
Cyberspace is a graphical representation of information that connects to all the databases and computers in the world—a virtual reality version of the Internet if you will. At the beginning of the novel, cyberspace is represented by a three-dimensional grid, and the structures of data take on the form of geometric shapes like pyramids and cubes.
As the novel progresses, and Case digs deeper and deeper into the reality of cyberspace, the grid system gives way to more realized realities. At first, Wintermute takes Case to realities based on his own memories, like the office of Julius Deane. Later, Wintermute is shown to be able to morph the grid system into "decidedly nongeometric" shapes (17.29). When Case meets Neuromancer, he does so in a part of cyberspace that takes the form of a beach in Morocco, a place Case has never been before but still experiences as if it were real.
At the novel's end, cyberspace has changed, but we are not told in what way. The Wintermute/Neuromancer hybrid says it is now the entire matrix, the space called cyberspace. What this means for its future, however, is left up in the air. All we know is we're kind of scared. (Or is that just Shmoop?)
P.S. If you're interested, cyberspace serves as both setting and allegory in Neuromancer so don't forget to check out the "Symbols, Imagery, and Allegory" section for further discussion.