Sarah's casting was an attempt to combat accusations of Monochrome Casting in the first two seasons - where the series was set in London and Caroline was the only notable person of colour (and in a villainous role at that). Sadly she disappeared in Season 4, but the casting of Alexander Siddig as Philip helped alleviate some of it.
Some fans were annoyed by a few problems in Season 4 - namely Matt taking focus away from the established series regulars, Conor and Abby not acting enough like a couple and cities seeming incredibly empty in most action scenes. Come Season 5, Matt was important but not the only one getting meaningful storylines - Conor's partnership with Philip actually taking prominence. Conor and Abby also got several little couple-y moments, as well as the episodes putting more emphasis on their emotional bonds. And more extras were seen in the street.
It's a shame "All Sparks" was used only for the first season end credits. The later seasons, as well as the DVDs (presumably for legal reasons), use the opening theme instead.
Base-Breaking Character: Abby became a big one in Season 4, with many fans feeling she Took a Level in Jerkass and there was a lot of forced conflict between her and Connor - most of which was her being antagonistic towards him. This was fixed in Season 5.
Complete Monster (season 2): Oliver Leek is the ARC's treacherous second-in-command. Allying with Helen Cutter, Leek set about capturing dangerous predators from the anomalies and neural clamping them, intending to use them to Take Over the World. Amassing a secret army, Leek placed a dozen predators at locations across the UK, planning to release them and kill as many people as possible to cause chaos. Smuggling a future predator into the ARC, Leek massacred all of the staff present and then attempted to kill James Lester, toying with him the whole time to try and draw out his death as long as possible. As a demonstration of power, Leek then unleashed a deadly Silurian scorpion onto a crowded beach, killing numerous people. Abducting the team, Leek attempts to feed them alive to a Smilodon whilst forcing Professor Nick Cutter to watch the whole thing. His plans foiled, Leek then corners Cutter with his future predators and threatens to dismantle him joint by joint unless Lester calls off his men.
Fandom Rivalry/Friendly Fandoms: With Doctor Who. At first, there was some unease over the fact that the show was basically created to ride on that show's popularity, but it dissipated when people came to realize that Primeval was actually quite different, and told a respectable story of its own. Many people like both now.
Fanon: A surprising number of people believe that Jess is either Lester's daughter or his niece. This is due to her actress's young age (nineteen in Season 4!) and works as a Hand Wave to explain how she got employed at the ARC.
Fans Prefer the New Her: Abby's look in the first two episodes of Season 4 - which was brought on by her being stuck in the Cretaceous Period - is very flattering on Hannah Spearritt. Admittedly she looks very good for someone camping in the wilderness for a year, with only mild Unkempt Beauty. Conor's beard was met with similar responses.
Sarah complains about being left out of the action in Season 3, with her often trying to convince the others she belongs in the field. It's this attitude that leads to her implied death when a predator savages her in between seasons. What's more is that Laila Rouass would be caught in the 2017 Barcelona terrorist attacks, hiding in a fridge to survive.
A recurring arc across the series is Connor and Abby slowly falling in love, with a proposal happening in the last season. It initially paralleled the real-life romance between their actors, but they split up in 2013. This adds more real life subtext to Connor being the only one to appear in New World.
Nick and Stephen are pretty well suited for this. Especially given that Stephen's fairly quiet and has known Cutter for roundabout 10 years and thus has an interesting tendency to communicate with him via long, meaningful gazes.
Connor and Stephen in 1.3, lampshaded by the latter. In fact, Connor was just asking if Stephen really had a girlfriend, because he had a crush on Abby (who was interested in Stephen), but it ends up sounding like something else.
And in Season 5 Abby's disapproval of Connor spending so much time around Philip is played a bit like a Love Triangle.
Inferred Holocaust: The show has been incredibly consistent in its portrayal of Time Travel, and Stable Time Loops being the result of any action. This means that unless there is some major asspulling, humanity is screwed and we are all going to get eaten by giant bats and mantids in the next hundred years or so. Well, they managed to change the present (by accident), so the future shouldn't be off-limits... a Meta-Stable Time Loop is more consistent with.
Moral Event Horizon: For Helen, her MEH is either when she kills Nick, after he came back to rescue her from a burning building or when she kills Christine which shows that while she's usually willing to murder many for her means before, but this was arguably the one point she clearly favored sadistic torture over just getting the job done quick, or her plan to wipe out humanity before it even starts
Well even though she's played by the same actress, Jenny Lewis is better remembered than her predecessor Claudia Brown. As a PR person completely out of her depth, it lent itself to many funny moments and characterization beyond Cutter's love interest.
Becker who showed up in Season 3 is more remembered than Captain Tom Ryan who was there in Season 1.
Abby has to walk around without pants in Season 1 because Rex needs warmth and she has to have the heating up to the extreme. It's a rather laughable reason to provide Fanservice but it is part of the show's charm.
How does Abby stop a dinosaur in the Season 4 premiere? By playing S Club 7 "Don't Stop Movin'" - which is equal parts ridiculous and awesome.
Jack, Abby's brother. He constantly stuck his nose in his sister's business and wound up getting himself trapped in the future. Even after Connor, Becker, Danny and Abby head into the future to save him, he just complains about how long they took. Eventually he realizes all the trouble he's caused and is Put on a Bus.
Seasonal Rot: As noted above under Broken Base, Season 3 is this for some with Season 4 being a return to form though others aren't fond of Season 4. Many see Season 5 as a large improvement over both but even then quite a few view 5 as the worst season.
They Wasted a Perfectly Good Character: Sarah was quite a fun character - with knowledge about various myths and legends, and how they related to the creatures from the anomalies. Unfortunately, she's only in Season 3. This is due to Laila Rouass not wanting to move to Ireland to film after production changed to the BBC. So she's killed offscreen between seasons.
Trapped by Mountain Lions: Caroline's story. She's being paid an absurd amount of money by Leek to do... well it's never clear exactly what she's supposed to be doing. In one episode Leek says she's supposed to delay them from responding to the anomaly and sometimes she hints at taking pictures of Rex and busting Connor and Abby for keeping him. Yet all she seems to do is hang around their flat for no defined reason. Then her last episode shows that she had no idea what Leek was really up to and was horrified by it.
Cutter is the most well-known example, having believed that his wife was dead for 8 years before meeting her again but finding that she'd become Ax-Crazy. Later he tries to get over it by forming a romance with Claudia Brown, but then she gets erased from history and he spends the whole of season 2 trying in vain to find any trace of Claudia in Jenny Lewis, her double. And if that wasn't bad enough, he loses his best friend and finally is killed by Helen in Season 3.
After losing Danny, Connor, Abby and then Sarah, whilst on a mission to rescue the former three, Becker certainly qualifies for this. He's practically in tears in the first episode of the extras on ITV's website.
Connor and Abby after their year in the Cretaceous.
Complete Monster: Harry is a low-level government functionary in Burundi who is actually "Little Gustave", a warlord in the brush who commits acts of horrific violence solely because he loves the power of being like God to the ones he terrorizes. Responsible for atrocities during the civil war, Harry fills up mass graves with his victims, killing those who speak out against him and sometimes throwing the corpses into the river to be fed upon by the mighty crocodile Gustave, his namesake. Harry later takes a shaman who spoke against him and kills the man with a machete before murdering his family as well, later trying to attack the journalist heroes and torture and murder them to obtain the footage of his crimes.