Adaptation Displacement: The play firmly established the popular image of Richard III as a crookbacked tyrant. To the extent where he's the only king of England to have his own fan club aimed at exposing this as a case of Artistic License History (spoofed in the first series of Blackadder, in which he really is a pleasant king who utters inverted versions of Shakespearean lines).
This was partly debunked / proven by the discovery of his remains, showing that he was not exactly hunchbacked but did had a bad case of scoliosis, resulting in the uneven shoulders seen in his portrait, meaning a case of both sides being somewhat correct.
Some productions like to depict Anne as being rather crooked and ambitious in her own right, and imply that she marries Richard not thanks to the power of his words and personality but because he puts her that much closer to getting a tiara again. This interpretation makes her own death rather karmic.
It is possible to get laughs into the scene where Richard seduces Anne. Watch.
"I am determined to prove a villain" - is Richard saying Then Let Me Be Evil, that he is choosing to be a murderous bastard out of pure spite; or, is he claiming he is Forced into Evil by fate and circumstance, that due to his condition he is unable to enjoy the "idle pleasures of these days" the way his contemporaries can, and that murdering his way to the top and wrecking havoc is the only pleasure he has left?
Complete Monster: Richard himself informs us early on that he is determined to prove a villain and ruin the day for everyone else. To that end, he seduces Anne Neville, whose noble husband he himself murdered, with every intent of discarding her later. He has his brother George, Duke of Clarence, sent to the Tower of London and murdered, drives his older brother King Edward IV into an early grave and has Edward's two young sons imprisoned in the Tower of London, before having them murdered. He poisons Anne herself, and begins having his allies killed. On the night before his battle with Henry Tudor, he is visited by the spirits of his victims, who tell him to despair and die. Richard is left alone, deserted by all, and at the end, he admits that even he has nothing but hatred for himself.
Draco in Leather Pants: Despite the negative intent of the play, a lot of fans and even Ricardians actually enjoy the play for Richard's sheer rambunctious energy. Harold Bloom even holds that Shakespeare's intent is that of an over-the-top parody of the official Tudor propaganda.
Evil Is Cool: Richard. Sure he might be a bastard but he owns it in such a way that he veers here and has style while doing it.
Evil Is Sexy: Richard's seduction of Lady Anne definitely qualifies. It's particularly marked in the Olivier film version, where Anne is all glazed-eyes and heaving bosom for Richard.
Funny Moments: Once Lady Anne leaves the stage after she's agreed to marry Richard, he muses "Was ever woman in this humor wooed? Was ever woman in this humor won?" as if even he can't believe he actually pulled that off.
Moral Event Horizon: Richard's murder of the Little Princes catapults him over the line and causes many of his allies to rebel against him.
Not only that, but the play itself rebels against him. Before the murder of the princes, Richard is dazzlingly evil and full of vitality. After their murder, he loses his vitality and his way with words. Taken from him just like *that!*.
Sequel Displacement: A fairly egregious example. The fact that this is a sequel / finale to Henry VI has been lost on both audiences and producers throughout history. Almost all of the characters in this play are originally introduced in that one, and watching Richard III by itself can actually result in some rather different interpretations of what's going on vs when the plays are watched in order as intended:
Richard himself seems more like a straight-up sociopath if one takes the play as a stand-alone, but as a sequel we see that he probably sees himself as the Only Sane Man in his family with both of his brothers being giant screw-ups in their own respective ways, endangering the dynasty while he was the one actually running the country properly, and growing increasingly frustrated with this state of affairs. Taken as a sequel character he's practically a Jerkass Woobie who undergoes Motive Decay from a loyal, faithful hardworking Yorkist to embittered asshole who thinks that neither Edward, nor Clarence, nor Elizabeth deserve or respect the throne as much as he does, so he might as well just take it for himself no matter how selfish his motivations have become.
Queen Elizabeth knowing that Richard "loves me not" and fearing what he'll do to her and her children makes more sense coming from Henry VI- Edward wasn't supposed to marry her in the first place, but to wed the sister-in-law of the King of France as a political marriage and as part of a peace treaty. He married Elizabeth solely because she caught his eye and she brought almost no political advantage whatsoever as she was just a regular aristocrat with limited standing, and France even goes to war with England over this and attempts to put the Lancasters back in power. Richard saw their marriage as a huge disaster and frankly as an insult to their whole family for Edward endangering their hard-fought, newly-formed dynasty over some woman he just met.
The tensions and mixed feelings everyone has towards Queen Margaret are explainable as well- in Henry VI, she is an absolutely horrible, downright evil person who abused, cheated on and bullied the husband she mourns for in this play, and killed the York brothers' father and younger brother in particularly nasty, humiliating fashion, and by an attack on their home after a peace treaty was signed no less; at the same time, Richard ruthlessly murdered her young son right in front of her without warning and this horrified the entire Yorkist faction and leads them to spare her life, which disgusts Richard further as he saw both of them as threats to the family and wanted to execute Margaret as well. That Richard topped it off by murdering her otherwise perfectly innocent husband who was beloved by everyone, even his enemies means that she is given more sympathy than she would otherwise deserve note Granted, none of them knew how much she bullied him, so they might have sympathised with her less had they known .
Narm: Several serious scenes are pretty hard to take seriously. For example, the scene where the members of King Edward's family rush to his bedside... followed by Richard of Shrewsbury on his little pedal car.
Richard III's death scene is also narmy, as he falls to his death with a massive smile on his face whilst waving to the camera with silly music playing in the background.
The scene where various messengers are rushing in to give Richard information on Buckingham and Richmond can look quite ridiculous when played.
Even the film's fans find that one time someone has a vision of Richard as a literal boar-faced monster ridiculous.
Woolseyism: McKellen modernized some of Shakespeare's dialogue, removing archaisms such as "thy" or "thou" and clarifying passages that relate to the Henry VI trilogy.