I should open this review by dispelling a common misconception. Shadowrun: Anarchy is not a 'rules light' game. Shadowrun: Anarchy has a pretty typical amount of rules for an RPG. It's only rules light in comparison to mainline Shadowrun, which is infamous for it's massive amount of rules. Anarchy is a streamlined Shadowrun experience, one that retains all of the core elements of the mainstream games, streamlines some of the more complicated rules systems, and adds in a bit of narrativist flair to appeal to a more contemporary gaming audience. It's meant to serve as an entry point for newcomers to Shadowrun, as well as an alternative system for people who want to run a quick pick up game, or simply prefer a less simulationist experience.
Shadowrun: Anarchy is designed to be a self contained introduction to the Shadowrun universe. Much of the early parts of the book are dedicated to establishing the the universe and it's more unique elements. There's a timeline, an overview of the various megacorps you'll be working for (or against), some short fiction, and a bunch of gorgeous artwork. The book also contains thirty premade PCs (complete with short backstories and character portraits), a gazeteer of the game's version of Seattle, and a massive set of "Contract Briefs" (adventure seeds). If you're looking to get into Shadowrun and can only afford one book, Anarchy will have everything you need to play.
The game runs on a variation of the Cue System used in the Valiant Universe Roleplaying Game. It's a narrativist system in which all of the players have a portion of control over the narrative, and are able to introduce new elements and plot twists as they see fit. Unlike traditional Shadowrun, the focus of the game is creating an exciting story, rather than overcoming the challenges of the run. By using Plot Points, players can freely help or hinder players in whatever ways they see fit, so long as the result is interesting. The result is a more unpredictable form of Shadowrun where the exact outcome of a given session can never be totally predicted (hence the Anarchy in the title).
Things are kept in check through Cues; small bits of description or narration that function somewhat like Fate aspects; players who are at a loss for what to do are encouraged to look at their Cues for suggestions, and Cues also serve to help dictate the tone of a given run.
However, Anarchy differs from Valiant Universe in that it maintains the traditional GM-Player relationship. There is a single person in charge of controlling the opposition and managing the runs. Interestingly, the GM doesn't have the power to directly help or hinder players; that responsibility falls solely on them. The result is a kind of hybrid Fiasco-Shadowrun game where players are encouraged to both help one another and screw each other over in pursuit of an interesting story. Of course, groups who aren't fond of this approach can also just play Anarchy like a traditional Shadowrun game. They lose out on some of the game's flavor, but they might end up with a more stable story.
Anarchy also differs from Valiant Universe in that Catalyst has gone out of its way to preserve all of Shadowrun's core mechanics. The "D6, Count Hits" resolution mechanic from Shadowrun 4e and 5e is still here, and most of the game's mechanics are reminiscent of (if not identical to) the mainline series, including attributes, skills, qualities, and condition tracks. If you're familiar with recent editions of Shadowrun, you'll feel right at home. If not, you'll still find character creation to be a lot more substantial than what you'd expect from a narrativist system.
The biggest change for Shadowrun veterans would be the introduction of Shadow Amps, which are meant to fill in for the various cyber limbs, magic spells, and matrix programs that a Shadowrunner needs to succeed. This is one of the more interesting aspects of the system, as the game goes with a sort of DIY approach; a character can only ever have six Shadow Amps, but can combine multiple effects into a single amp, resulting in custom spells or tricked out cyberware in more creative players. The game also encourages GMs and players to design Amps from the ground up, though there's no clear guide on how to do so. The classic Magic/Resonance/Cyber divide is still present as well. Being Awakened (or a Technomancer) costs two amp points (but not an amp slot), and most Cyberware reduces your Essence score (which weakens your ability to do magic).
Shadow Amps have a lot of potential; they feel like an immensely hackable system, and a savvy GM could use them to create things that mainline Shadowrun cannot or does not support. While the game doesn't include any of Shadowrun's non-traditional concepts (like AIs, Free Spirits, Awakened Critters, and Cyberzombies) the Shadow Amp system is flexible enough that introducing those elements to the system would require only a minimal amount of work.
Weapons and gear have also been abstracted. Nuyen and Lifestyle Costs are nowhere to be found (except as a narrative element). Weapons, Gear, and Contacts are now bought and upgraded through Karma (the game's version of experience points). Weapons still generally resemble their Shadowrun counterparts (albeit with fewer modifiers), while Gear has been abstracted to its purely narrative purpose. Riggers may mourn the lost of having ten thousand modification options for their van, but others might revel in being freed from the shackles of bookkeeping. It really depends on the group.
Finally, some of the game's subsystems have also been simplified. The Matrix (a notoriously complicated system) has been simplified to requiring only a single roll to gain control of a given matrix object, with things like alarms and the Overwatch Score being relegated to Narration twists. Matrix combat is still a thing, and functions mostly identically to it's mainline counterpart. There are also optional rules designed to make the matrix more similar to it's 5e counterpart for players who prefer a more substantial decking experience.
Magic operates much like it's Shadowrun counterpart, though there is no Drain to speak of. A magician can cast a given spell indefinitely without fear of injury; a fair trade for the six spell limitation that all players are now restricted by. Sustaining spells is also much more simple, though there are still (suggested) penalties for doing so if the spell is particularly complex, or being held for a long time. Spirit summoning is also relatively similar, although players can no longer bind spirits, nor can they have multiple spirits summoned at the same time.
Technomancy has undergone similar tweaks; Fade is completely gone, but players can only know six Complex Forms and have one sprite compiled at a given time.
The Rigging rules are also mostly functional, although the book only provides three example drone templates (and zero vehicle templates) buried in the NPC section, which leaves players and GMs to basically improvise if a rigging focused character is desired.
In fact, Improvisation is Shadowrun: Anarchy's biggest weakness (despite also being its biggest strength). While the freeform, narrativist style allows for high speed gameplay and a far more flexible experience than what 5e could ever give, there are many situations where the book simply shrugs and lets the GM and players figure things out on their own. Riggers are forced to try and puzzle out how to create drones and vehicles using the NPCs as a template. GMs are forced to try and reverse engineer the rules for creating Shadow Amps. The Non-Player character section is pitifully short, and completely lacks illustrations or descriptive text; it's just several pages of statblocks with vague titles like "Corporate Suit" or "Bug Queen". These are all relatively easily resolved issues (even a few sentences would be sufficent in each case), so it's baffling to see Catalyst fall short like this.
So, is Shadowrun: Anarchy for you?
If you're new to Shadowrun and want an easy way to run a game for your friends, this is absolutely a good choice. Just be sure your group is willing to play along with the Cue System for an ideal experience.
If you're a Shadowrun vet, I'd rank it as a maybe. It's similar enough that you'll feel at home with the mechanics, but the spirit of the rules might throw off certain groups. It's definitely worth a look if you've ever been frustrated with the mainline systems limitations and complexities, but if you love Shadowrun for its complexity, then Anarchy isn't for you.
However, if your main love of Shadowrun is for it's unique setting, and you just want a chance to experience it without all that fragging bookkeeping getting in your way, then Anarchy is definitely worth the price of entry.