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The basic idea is that Albert Einstein had proposed a secret community for the country's greatest minds to do their scientific research in seclusion, I guess. Thus was founded the town of Eureka, somewhere in the Pacific Northwest, though few people know of its existence. For the past 50 years or so, our greatest innovations have been created in Eureka, though there are also occasionally accidents when experiments go awry, which may threaten the continued existence of the world, or even the space-time continuum itself. Though it's not always quite that drastic, there does seem to be some kind of bizarre problem, every week.
Anyway, one day a U.S. Marshal named Jack Carter is transporting a teenaged girl, Zoe, to Los Angeles; she's apparently some kind of criminal, though it's soon revealed that she's his daughter. She doesn't get along with him well, because she blames his dedication to his career on his divorce from her mother. Well, they have an accident and get stuck in Eureka, at least the portion of it that looks normal enough. It will take a few days for the local mechanic, Henry Deacon (a former space shuttle engineer) to fix Jack's car. (Over the course of the series, Jack and Henry become best friends, and Henry often helps out with Jack's investigations. But I'm getting ahead of myself.) Meanwhile, Jack leaves Zoe in the local jail, to be looked after by Deputy Jo Lupo (an ex-Army Ranger who really likes big guns). Jack stays at a bed & breakfast run by Beverly Barlowe (a psychotherapist whose clients include not only the brilliant scientists of Eureka, but all sorts of important people, from celebrities to heads of state).
At first glance, the town might simply seem particularly quirky, but it soon goes beyond quirky to downright eccentric, and then bizarre. When a boy goes missing after an accident in his parents' RV, Jack butts his way into the search, being conducted by Sheriff Bill Cobb, as well as an agent from the Department of Defense, Allison Blake (actually a liason between Eureka and the Pentagon). Jack is the first to find the boy. The accident, which had destroyed the back end of the RV, had involved a vortex created by one of his father's experiments. And these vortices continue showing up around town. One of them nearly kills Cobb, and while he's incapacitated, Blake recruits Jack to help investigate the phenomenon. She makes him sign a non-disclosure agreement (if he reveals anything he sees, he will be charged with treason), and takes him to the town's hidden Advanced Research Facility. (I loved it when Jack asked, "Where are you taking me, Area 51?" and Blake responded "Please. They wish they had our security.") He's still not allowed to see Section 5, where secret military projects are conducted. Eventually, Jack ends up saving the day... or at least figuring out who will be able to save the day. The vortices are stopped, and Jack heads back to L.A. But then, he's given a promotion: he's going to be the new sheriff of Eureka, as Cobb has retired.
Well, I should probably mention some other characters. First of all, the Advanced Research Facility is run by an organization called Global Dynamics. In the pilot, it seemed to me that the head of Global was Warren King, but I may have been wrong about that. Or, maybe he was just replaced, as I've never seen him again in the series, but I'm not sure what exactly has become of him. Anyway, the new head of GD is Nathan Stark, who is Allison Blake's soon-to-be ex-husband. He and Carter really don't get along, and it does seem like something more than a professional relationship might develop between Carter and Allison. Oh, and I should mention that there's also some mysterious object contained in Section 5 that Stark wants to learn the origins of. Most people don't seem to even know it exists.... There's also a scientist named Douglas Fargo who seemed to be a lackey for Warren King, and now for Stark. Oh, and Jack and Zoe's new home in Eureka has an artificial intelligence running everything. It's called S.A.R.A.H. (Self Actuated Residential Automated Habitat), which was programmed by Fargo, and speaks with his voice (but feminized). And there's a "biological containment specialist" named Jim Taggart (played by Matt Frewer, with an Australian accent), who's also a cryptozoologist and veterinarian. And there's Vincent, who runs a local diner called Cafe Diem. And Spencer, who, I forget what he does, but he sometimes hangs out with Fargo and Vincent (though I didn't see him in the latter seasons, I'm not sure what became of him). Allison and Nathan have a son named Kevin, whose role will be somewhat different in season 4. In season two, a former hacker named Zane Donovan comes to Eureka. And in season three, the DOD sends a corporate "fixer" to Eureka, named Eva Thorne. Also in season 3, we meet a scientist named Tess Fontana, who eventually begins a relationship with Jack (for awhile). And I suppose I should mention General Mansfield, a military guy who seems to have ultimate authority over everyone and everything in Eureka, though he only shows up occasionally. In season 3, an android sheriff named Andy was introduced, briefly, but he'd play more of a role starting in season 4, as a deputy. And there may be other characters I wanted to mention, I forget.
Actually, I'm sure there's a great deal more I should probably say, about some of the changes that happen throughout the series. But for now I'm not going to, both because it's too spoilery, and because I just don't remember exactly how everything happened. But there are definitely some changes. Some characters leave, new ones show up, old ones' positions get switched around, and whatnot. And while I think of this as being just as much a quirky show as sci-fi, it is important to note that some stuff happens that could only happen in science fiction, which leads to a lot of changes in circumstances, and... changes in the way certain characters think, and what they do, even whether they're basically bad or good. Or whatever, I dunno. It's complicated. Still, the more things change, the more they stay the same, plotwise. (You know, except for the things that change.) If you want to know more, check out some links, okay? Meanwhile... all I can really say is I like the show well enough, but I don't love it. It's just some good, simple, complicated fun. And it's amusing. And weird. And stuff.
Um. But I absolutely have to mention season 4... is different. In the season premiere, Jack, Allison, Henry, Jo, and Fargo unexpectedly get transported to 1947, around the time Eureka was founded. They meet Dr. Trevor Grant, who ends up being transported with them back to the present (where he calls himself Charles Grant, ostensibly a historian). They find that there are some differences from the history they remembered, though I don't want to say exactly how things have changed (aside from the addition of a character named Dr. Grace Monroe, who becomes very important to Henry). But they do have to keep secret the fact that... they time travelled, and all. Grant leaves halfway through the season. In the second half of the season, a scientist named Dr. Holly Marten (played by the always adorkable Felicia Day) comes to Eureka to help with the plotline that dominates the rest of the season: preparations for the launch of a mission to Titan (one of Saturn's moons), on a new ship called Astraeus. Holly will become a love interest for Fargo, though there's some competition between him and Dr. Isaac Parrish (played by Wil Wheaton, which is cool because he'd also been on Day's webseries The Guild). Of course, there are standard stories of the week while all that goes on, as well as ties to things that had happened in earlier seasons (including the return of Beverly Barlowe, who at some point had become a nemesis, and has only rarely been seen since... meh, I don't even remember when she left Eureka). And of course those who were new to this timeline still had to deal with things not being quite as they remembered, most notably the relationship between Jo and Zane. Also, between seasons 4 and 5 there was a holiday episode which doesn't take place in the regular continuity.
Anyway, at the end of season 4, the Astraeus launches prematurely, with not only the crew on board (in suspended animation), but also Allison, who wasn't supposed to be there. At the start of season 5 (the final season), the ship returns, but it seems four years have passed on Earth while they were gone. The crew hasn't experienced any of that time, so they have to adjust to all the changes that have happened while they were gone. However... things soon turn out not to be what they seem. I don't want to spoil exactly what was really going on. But I'll say Beverly Barlowe plays an important role in the season, to some extent. And Holly becomes a major focus of an ongoing plotline. And of course, as always, there are regular crises to deal with each week, on top of the major story arcs. That's all I want to say about the season's plot, but I will say that the series finale was quite satisfying. I should also say that the series as a whole steadily grew on me over the years. I still don't think I'd say I "loved" it, but... I definitely came to appreciate it a great deal, and really liked all the characters.
In season 1, there was a series of webisodes called "Hide and Seek," which I don't think I watched, but I may try to do so, someday.