Adorkable: Bernard and his "Gosh." and "Crikey." and occasionally speaking at length about etymology.
Base-Breaking Character: Dorothy Wainwright from Yes, Prime Minister; fandom is split as to whether she was the strong, politically minded female character that Yes, Minister was sorely missing (Annie, while a fairly strong character overall, was more of an everywoman), or a Creator's Pet who is constantly shown to be right about everything, and is able to quickly reduce senior civil servants to babbling idiots in a way that even the likes of Ludovic Kennedy couldn't manage. Notably Claire Sutton, Dorothy's spiritual successor in the play and 2013 series is considerably less moral and while still brainy makes one or two huge mistakes of her own.
Germans Love David Hasselhoff: The series was so popular in the Czech Republic that one theatre ordered a play to be written based on the show's premise with the beloved characters. The roles were performed by those actors who dubbed them in the series. The play was a smashing hit and it is still on. In The New '10s.
In a show written by two people who saw how British government works from the inside, Hacker's suspicious and often hostile attitude to Europe speaks volumes about how so many senior politicians in both parties were already predisposed towards europhobia and Brexit, even several decades before the event. Admittedly some of it is so over-the-top that it can be separated from real life (Humphrey saying the French are more dangerous than the Russians and Hacker actually buying it—for a moment), but things like Humphrey matter-of-factly stating that it's Britain's job to keep Europe disorganized ring a little bit too true in the modern day.
Hacker's plan to become Prime Minister through lying about European standards on food and then declaring a victory over Brussels by convincing them to change their plans to rename sausages in Britain can come across as unpleasantly similar to Brexit and Boris Johnson's rise to Prime Minister, considering that Boris Johnson while a journalist helped to foster Euroscepticism by writing blatantly untrue stories about European standards on bananas, etc. One wonders if he was taking notes from the series. This is particularly glaring, given Hacker's tendency to slip into the cadences of Winston Churchill whenever he's feeling especially bombastic: Johnson wrote a book about Churchill, which was widely reviewed as being an extended exercise in making Churchill look like himself.
All the references to the show Jim'll Fix It, after the revelation of Jimmy Savile's horrendous sex crimes.
In "The Moral Dimension," Bernard tells Hacker that he's got a call from "Mr. Haig" in the communications room they set up in the Qumrani royal palace. While Bernard's message was actually to advise Hacker on the availability of Haig whiskey illicitly smuggled into the palace, it carries an extra layer of hilarity when you consider that the role of Hacker was taken over by David Haig for the stage play and 2013 series. There is also the fact that the Foreign Secretary in the early 2010s was called William Hague.
Far more likely to allude to Alexander Haig, who was at the time of writing the US Secretary of State
Multiple times the PM's special adviser Sir Mark Spencer appears, the name being a joke on the chairman of Marks and Spencer, Sir Derek Rayner, a Government adviser at the time. In 2010 a Mark Spencer entered Parliament who has held some Cabinet roles.
In "The Smoke Screen", Dr. Thorne's proposals to attack smoking by banning all advertising (even at point of purchase), drastically increasing the tax on cigarettes and instituting a ban on smoking in public places is viewed by the other characters as impractically radical and impossible to implement. Pretty much all of his proposals have (in some way, shape or form) since become real-world government policy in Britain and many other places, albeit they've generally been phased in gradually rather than in the large, sweeping manner that Thorne was proposing.
In "Doing the Honours" Bernard tells Hacker a joke about how the post-nominal "GCMG" stands for "God Calls Me God". In the novelization Bernard rises to Head of the Home Civil Service and receives the even higher honour "GCB" - meaning he outranks the people that God calls God.
In "Party Games", Sir Humphrey and the government chief whip agree that neither of the two front-runners to succeed the retiring Prime Minister is desirable, with the chief whip mentioning that either of them would split the party, and that the next Prime Minister needs to be someone quieter and more moderate. This uncannily predicts the situation in which John Major came to succeed Margaret Thatcher six years later, with Major getting the top job largely because fellow contenders Michael Heseltine and (to a lesser extent) Douglas Hurd were considered too controversial.
No Problem with Licensed Games: The Yes, Prime Minister computer game from the 80s plays essentially as a choose-your-own-adventure game where you navigate through one week of meetings as Hacker and make policy decisions by picking your responses from a list. As one might expect, a game like this lives or dies on the strength of its writing, which fortunately is very good. So much so that the back of the box features glowing endorsements from Antony Jay and Jonathan Lynn themselves (who had no direct involvement with the game).
One-Scene Wonder: Graeme Garden as Commander Forrest in "The Death List", matter-of-factly reeling off all the possible ways Hacker could be assassinated.
Frank Weisel. Even leaving aside the thematic reasons as to why he didn't work as a character, most fans found him extremely irritating, and he didn't add much to episodes beyond giving exposition (which Annie or Bernard would instead give in later episodes) and a reason for Sir Humphrey to make a Weasel/Weisel joke.
Hacker's daughter Lucy, while not as unpopular as Weisel (mostly because she only appeared in one episode, as opposed to the entire first season), isn't particularly liked by fans either, due to being a two-dimensional stereotype of a student activist.
Tear Jerker: When Bernard makes a serious (if well-intentioned) mistake in 'The Moral Dimension', he seeks Sir Humphrey's advice and is told he must inform the minister. Bernard admits to Jim what he has done, and then they are informed that a journalist is outside, at which point Bernard gives a small cough that could easily be mistaken for a sob. He asks what Jim will do, and when he hears that Jim has no choice but to tell the truth, Bernard is visibly upset. He continues to protest, weakly, but nothing can be done. Fortunately for Bernard, this becomes heartwarming when Sir Humphrey steps in and saves his skin.
Unintentionally Unsympathetic: To some viewers, Sarah Harrison from the episode "Equal Opportunities." While the "The Reason You Suck" Speech that she gives to both Sir Humphrey and Hacker at the end of the episode is 100% deserved by both men — the former for his outright sexism, and the latter for trying to get Sarah appointed to a position she had no interest in purely to make himself look progressive — no-one points out that by quitting the civil service just because there are so few other women working there, she's helping create a "chicken and egg" scenario whereby other women won't apply to be civil servants for the exact same reason, and if anything will be further put off by a woman as accomplished as Sarah seemingly admitting defeat and quitting.
Values Resonance: It's surprising how relevant some of the political issues explored in the show (though by no means all) still hold true in later decades. For example, one episode deals with upgrading the British nuclear deterrent to Trident, in recent years the issue has been replacing Trident; despite being set in the Cold War, it's portrayed as just as ridiculously pointless as many think it to be now. Other issues including government waste, data-gathering and privacy concerns, Britain's place in Europe... And, whilst it was said to draw more by way of examples from the pre-Thatcher era than its own time (Thatcher taking a much harder line with the Civil Service than Jim Hacker ever dared), it remained a big hit with the then-PM and her cabinet.
Harsher in Hindsight: The plot of the stage play involves an oil-rich central Asian state negotiating a pipeline deal with the government. The deal is jeopardised by the state's foreign secretary asking the British government to supply him with an underage girl with whom he can have sex. In the light of the Me Too movement and the trafficking of Jeffrey Epstein, this comes across as exploitative rather than sleazy. The 2013 TV adaptation isn't quite as bad in that regard, with the foreign secretary instead having designs on Claire Sutton, who is willing to go through with it, with the main sticking point being her holding out for a promotion and a big cash bonus; it still comes across pretty tasteless, however, especially since Mad Men had more realistically explored implications of such a scheme just a few years beforehand.
Sequelitis: To say that the 2013 Yes, Prime Minister series wasn't as well-received as its forerunner would be a gigantic understatement. Common criticisms included:
Flanderization of all three major characters. Hacker became far more bad-tempered and ineffectual than the original version ever was, Sir Humphrey became so corrupt and such a megalomaniac that he seemed to become similar to Alan B'stard, while Bernard seemed to have degenerated from being idealistic and pedantic to being an outright Manchild.
Padding: The series started out as a stage play and retained broadly the same storyline, resulting in a lot of clumsy exposition dumps. In particular, the entirety of the third episode basically consists of just two scenes; firstly Sir Humphrey trying to manipulate Hacker for fraudulent expense claims, and then Hacker trying to do the same to Sir Humphrey for illegal usage of government credit cards.
Unintentional Period Piece: While the original series generally manages to avoid this for the most part, the 2013 series experiences it far more thanks to it being clearly based on the UK political situation in the early-mid 2010s, with the country being ruled by a coalition government, the government being beset by expenses scandals, and there being talk of closer links with the European Union to offset the damage caused by the Great Recession (in real life, of course, the complete opposite happened). Though one area where it does come across as oddly prescient is by depicting Hacker's coalition partners as being the Scottish National Party, presaging the complete collapse of the Liberal Democrats (who were the junior partners in the 2010-2015 Conservative-led coalition government) at the 2015 general election, and the SNP rising to become the third-largest party.