In the 90s and early 2000s my manga reading was mostly through borrowing actual volumes from friends. In the mid to late 2000s, the friends I borrowed from I saw less often, so my manga reading often happened online via scanlation groups and now-defunct websites like OneManga or MangaStream. Last year as preparation for an anime/manga themed quiz night, I reviewed a bunch of manga I read in the 2000s, I thought I'd write about them. I only cover here series that are already done. (There are some series I had read during this era that are still ongoing and are covered in my 2023 Manga Update!)
Full Metal Alchemist

Easily my favorite manga (and anime) of this era. Great characters, plots, worldbuilding and an epic ending. Both of the anime series featured great animation. Roy Mustang remains one of my favorite characters to this day (maybe helped in no small part by his name). It is also very well-known and super popular, so I don't feel the need to elaborate on it.
I think I started reading the manga around 2005 or 2006 (it started in 2003) and followed it until it ended in 2010. I watched both anime series, with Brotherhood being the superior one of course. I have not seen the live-action version, reviews weren't great.
Liar Game

Liar Game is a kind of "puzzle box" type of story where the characters are thrust into a competition where they have to overcome complex and devious challenges that requires them to think many steps ahead of the other participants. The lead characters are a naive college student thrust into this sequence of challenges and a clever con man who she asks for help. The stakes are high - it is heavily implied that those who lose the game drown in debt and may even be sold off to slavery. It's the type of story where the game presents a series of convoluted puzzle traps that the competitors must think their way out of, and the face more clever opponents as the series goes on. The "con man" main character is a seeming super genius who thinks many moves ahead and anticipates a lot of the counter-plays from the other participants and the Liar Game organization and they somehow manage to barely squeeze through each challenge.
I remember bingeing most of what was available for this series in the late 2000s. Last year I found out I stopped reading around Dec 2011 and never got around to finishing it, so I decided to read the remaining volumes. I think roughly a third of the series was still remaining. I kind of regretted it, as the series ended in a terrible and anticlimactic and completely unsatsifying way. The peak of the series was probably around the time I stopped reading it in 2011, during the "Musical Chairs" arc.
Death Note
Another well-known manga, this is another one that features dueling super-smart characters in the form of protagonist Light Yagami and his nemesis (and actual good guy) L. The first half of the manga is much better than the second because it focuses on Light and L's rivalry and them constantly trying to outsmart each other in their cat and mouse game of murderer and detective. The story changed in the second half and became much too convoluted in my opionion.
I think one of the reasons for this manga's popularity was just the fantasy aspect of the Death Note itself as it's so easy to imagine how convenient having such a thing would be. Like when the 2010 hostage crisis happened over here, I jokingly posted on social media about how we wouldn't have these problems if we had a Death Note! I also surveyed my friends to ask how they'd handle having a Death Note!
The series ran from 2003 to 2006, and I believe I read it until the end, but I don't actually remember what the ending was. I actually have a physical copy of volume 2 somewhere around here.
Bakuman

Bakuman was the next manga by the creative team that did Death Note and the tone could not be any more different. Bakuman follows a writer/artist pair who grew up together and team up to fulfill their dream of breaking into the manga business. I have no idea how accurate the portrayed mangaka experience is; I suspect it is sanitized a bit to tone down how overworked such people are, and even then the characters still work way too hard.
At a certain point in the story, the two mangaka had a problem with their current series because it was based around a rivalry around two strong characters and they were worried that either the rivalry would go on for too long or if they changed it up, the series would become weaker. This was an obvious reference to their previous work in Death Note!!
I really enjoyed this one. This series wasn't very long, but it had a great, clean art style and engaging storylines. I read the series from Nov 2009 to its eventual ending in Apr 2012.
Claymore
Claymore was a fantasy action series following the adventures of female warriors called Claymores tasked with slaying demons who preyed upon humans in a certain continent. This series was dark and had great worldbuilding, but I always found myself annoyed by the art because it was the sort of thing where the demon enemies were always in various massive shapes that made a lot of the battles hard to follow. Also a lot of the Claymores looked alike, adding to my confusion. It was also a monthly; the long time between chapters meant I constantly kept forgetting what was going on. The series ran from 2006 to 2015, and from my notes I was still reading it in 2009, but I suspect I stopped reading it shortly after.
Cromartie High School

Probably the least well-known of the manga I'm discussing here today. I have a specific friend group that loved this one. Cromartie was a high school known for its high population of delinquents. Aside from the typical high school tough guy hijinx, it also had a strong fantasy element. Without explanation, the students included a gorilla (he couldn't talk or anything like that, he's just an actual gorilla) and also a robot shaped like a giant oil drum. It had a lot of over-the-top absurdist and often nonsensical comedy that we found hilarious.
The series ran from 2000 to 2006 and I think I was reading it around 2007 and I don't know whether I've read all of it, but it's the kind of series where that doesn't really matter. Hm, maybe I'll look for a copy!
That's It!
There's a bunch of other manga I read during this era, but I think the above are the ones that stuck with me for at least a few years.