The series consists of (in order): Nine Princes in Amber, The Guns of Avalon, Sign of the Unicorn, The Hand of Oberon, and The Courts of Chaos.
I think this is my third time reading through this series, the last being in 2013. I first read them when I was quite young: I hadn't yet become picky about the quality of the writing, caring only that the story was entertaining. In my 2013 re-read, I wept at the sloppy prose. But the story is so inventive and entertaining that I had no trouble getting through the five books. I was even more painfully aware of the prose this time, and I wondered if he'd done it on purpose. It's worst in the first two books, and Zelazny is capable of far better writing: maybe he wrote this way because this is how he imagines Corwin (our first-person narrator for the entire series) speaks. And because it was the Seventies.
"Why?" I asked him, and the sound of my own voice was strange to me. There was silence. He emptied his pipe. He refilled it. He relit it. He puffed it. There was more silence. Then, "I don't know," he said. ...
It's deliberately ponderous and awkward. No idea why.
But the plot is fast-paced and fascinating, and more than interesting enough to overcome the bad prose (which is saying something because it's really, consistently bad prose).
Our hero awakes with amnesia, but over the course of the first book discovers that he's a prince of Amber, and he and his kin can "walk in Shadows," which is to say they're able to traverse a multiverse of worlds, all of which are "shadows" of Amber, the perfect world (although it's home to some nasty politics among the nine princes).
Of the five books, the final one is the weakest: Zelazny has to create an appropriate grand finale, and that includes ... well, the destruction of reality. It gets awfully metaphysical and philosophical, which isn't much of a treat in that lousy prose (although it's improved since the first couple books). And it involves a lot of "hellrides" (fast switching between multiple worlds which often produces unpleasant side effects). The problem isn't the hellrides, per se, but rather the way he writes them with pages and pages of disjoint prose, suggestions of images. There's way too much of it in one book.
Despite these complaints, the series as whole is a blast, an epic fantasy series I will still - despite the prose - recommend to any fan of the genre.
There are five more Amber books, written from 1985 through 1991. I only managed to read the first in that set (years ago): it seemed clear that it was structured like a video game, with "the bad guys" and "the good guys" alternately getting bigger and better power-ups. With my view on the subject confirmed by other reviewers who read the rest of the series, I'd recommend avoiding the second series.