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Caution: potential vague spoilers maybe.
Volume One: Genesis
Okay. So, there are various people around the world who discover they have developed super powers. There's a Japanese guy named Hiro Nakamura (Masi Oka), who can bend the space/time continuum, or whatever... which means he can not only affect the flow of time or travel into the past or future, but also teleport. And um, he's like a fan of Star Trek and comic books and stuff. I had a feeling from the very first episode that he could become my favorite character, and he did. In fact Hiro seems to have become many fans' favorite character, and the media obviously loves him.... There's also a high school cheerleader in Texas named Claire Bennet (Hayden Panettiere), who seems to be indestructible, and quickly heals any injury. And in New York, there's a heroin addict named Isaac Mendez who can paint things that will happen in the future- at first, this only happened when he was high, but eventually he learned to use the power without using drugs. There's a woman named Simone Deveaux, an art dealer who was involved with Isaac, but left him because... well, both because he was an addict and because he believes he can paint the future. She doesn't seem to have any powers, herself. Coincidentally, her father is being treated by a hospice nurse named Peter Petrelli (Milo Ventimiglia), who believes he can fly, and Simone starts dating him. Though at the time I don't think she's aware of this belief of his. But he will also believe in Isaac's power. Anyway, Peter's older brother, Nathan (Adrian Pasdar), is running for Congress, and he can fly, but he doesn't want anyone to know about it. He sees Peter as a hindrance to his campaign. As for Peter... he has a gift, but it will turn out to be something much bigger than flight.
In Las Vegas, there's a stripper named Niki Sanders, who has a mirror image (literally) that seems to take over her body sometimes. It's more than just an (evil) split personality, this alter ego (called Jessica) seems to be super strong. Also, Niki has a son named Micah, who will eventually turn out to have a power of his own. As does his father, D.L. Hawkins, who at the start of the series is in jail, having been framed for the theft of $2 million and the murder of his crew. But he escapes prison when he finds he can phase through solid objects. And there's an Indian genetics professor named Mohinder Suresh (Sendhil Ramamurthy) who's come to New York to follow up on his father Chandra's work, studying the possibilities of human evolution (Mohinder often narrates the show). He meets a woman named Eden McCain (Nora Zehetner), who claims to have been his father's neighbor, and helps Mohinder out in his investigations for awhile.
Well, everyone's connected somehow, or will be. Hiro teleports from Tokyo to New York, and finds Isaac dead. Later, he witnesses an atomic explosion, and teleports back to Tokyo, where it turns out that his trip to New York took place 5 weeks in the future. So, after convincing his friend Ando Masahashi that he really does have this power, the two of them take off on a trip to New York, hoping to prevent both Isaac's death and the nuclear explosion. Ando speaks good English, though Hiro only speaks a little. But he gets better as the season progresses, and a version of him from five years in the future appears to Peter on the subway with a message: "Save the cheerleader, save the world." This would become the show's major catchphrase, at least for the first part of the first season. Of course, the entire season is building toward the explosion Hiro witnessed, which may have been caused by one of three people....
There's also a cop in L.A. named Matt Parkman, who develops the ability to read minds. This gets him involved in a case being handled by the FBI, working with an agent named Audrey Hanson, who's tracking a serial killer known as Sylar (Zachary Quinto). Apparently Chandra Suresh was aware of Sylar, who seems to have powers of his own, and is killing other people with such powers, so he can take their powers for himself. There's also a mysterious organization that includes Claire Bennet's adoptive father (who's come to be known to fans as "Horn-Rimmed Glasses" or HRG, which seems to me rather reminiscent of Cigarette-Smoking Man from The X-Files, though we eventually learn his actual first name). And Claire has no idea that he's involved in this- at first. Some people with powers work for the same organization, including a man called "the Haitian," who can remove people's memories as well as apparently block their powers. But he has some secrets he keeps from Bennet, including other loyalties. As for Bennet himself, it's unclear whether he's a bad guy or a good guy... except that he wants to protect his daughter. There's also a mobster in Las Vegas called Linderman, who is often alluded to, and has ties to various characters, though we don't actually see him until near the end of the first season. But he's very important.
We eventually meet a man named Ted Sprague, whose wife died of radiation poisoning because it turns out he has nuclear powers, which he has trouble controlling. It seems he may be responsible for the explosion Hiro witnessed in the future, though Peter eventually begins to think he himself may be responsible. And of course, the third possibility is Sylar. Well, we'll continue to meet various other characters, both with and without powers. And there are family members of various people who will be important, including Nathan and Peter's mother, Angela Petrelli, and Hiro's father, Kaito Nakamura (played by George Takei!) Throughout the series, plenty of new characters are added, and occasionally characters die.
There's so much just in the first season that I have avoided mentioning, in the way of both plot and characters. And the show actually had four seasons. I don't really want to say much about anything after the first season, though you should know that a lot of fans didn't like much of anything beyond the first season. Personally I liked the whole series, and some of the stuff that other people specifically disliked, I actually liked. But even I will admit the first season was the best, though I did like some of the characters who weren't introduced until later...
Volume Two: Generations
Hiro spent some time hundreds of years in Japan's past, where he met a girl named Yaeko (Eriko Tamura). He also met a guy called Takezo Kensei (who his father used to tell him stories about, but who turned out not to be what he expected at all). Back in the present, there's a woman named Elle Bishop (Kristen Bell), who has electric powers and is the adopted daughter of Bob Bishop (Stephen Tobolowsky), who works for the same company as HRG. And there was a girl named Monica Dawson, who was Micah Sanders's cousin. She has the power of muscle mimicking (which is like... she gets muscle memory just by watching actions, rather than performing them... so once she's seen any action, she can perform it, automatically). There's also a guy named Adam Monroe (David Anders), who... well, I don't want to spoil anything about him. ...I'm leaving out characters and don't really want to divulge any of the season's major plot, but I will say the season was cut short because of the writer's strike in Hollywood that year (2007).
Volume Three: Villains
Unlike the first two seasons, which each contained a single volume (or story arc), season three had two volumes. In the first, there were a bunch of villains that escaped from the Company. Meh, I don't want to talk about that. There was also a girl named Daphne Millbrook, whom I definitely liked. She had super-speed. And Matt Parkman was interested in her, because of some vision he had, provided by... some African dude. (But I was more a fan of Daphne's encounters with Hiro.) There was also someone named Tracy Strauss, who looked just like Niki Sanders, for a reason I won't go into. Also at one point there's an eclipse during which everyone loses their powers (which kind of reminded me of "Day of Black Sun" from Avatar: The Last Airbender). Also we meet Arthur Petrelli, who was Nathan and Peter's father (who of course has a power of his own). Once again, I'm leaving out tons of characters and plot developments.
Volume Four: Fugitives
In the second half of season three, the U.S. government establishes a group to hunt down people with powers. One of the main people in this group is a guy named Danko (Željko Ivanek), who works with HRG (the Company he previously worked for no longer exists, at this point). And again I don't want to say much about the story, but I will say we learn a bit about stuff that happened several decades earlier.
Volume Five: Redemption (season 4)
This season there's a carnival full of people with powers, led by someone named Samuel Sullivan (Robert Knepper), who has big plans. Meanwhile, Claire begins attending college, where she meets a girl named Gretchen Berg (Madeline Zima). Also we continue to learn more about the past. And... meh, I don't want to say any more.
Throughout the series, there were online comics at NBC's website, as well as several series of webisodes. Most of the webseries I wasn't all that into, but what I saw of them was okay. Though I will say the comics tended to be more interesting to me. Some of them were hard to follow and I didn't like that much, while other installments of the comic I thought were quite good. Some of them provided some very interesting information that genuinely enhanced my understanding and appreciation of certain plot elements of the actual TV series. Anyways... I dunno what else to tell you about the show. Yeah, um... parts of it weren't that great, and parts of it really were great. Other parts were just sort of good. Tons of characters, tons of mythology, tons of... everything. I'm not sure it was always consistent; certainly it could be a bit confusing at times. A bit of a mess. But... for the most part, I definitely liked it. I think I would have liked it to last at least one more season, but... meh. I don't mind that much that it ended where it did, I guess.
Well... it ended in 2010, but in 2015 there's a 13-episode sequel series called Heroes Reborn.