These are what we call the 'YMMV items.' Things that some people find in this work. We call them 'your mileage might vary' because not everyone sees these things in the same way. This starts discussions in the trope lists, a thing we don't want. Please use the discussion page if you'd like to discuss any of these items.
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 by the discovery of his remains, showing that he was not hunchbacked but simply had a bad case of scoliosis, resulting in the uneven shoulders seen in his portrait.
Alternative Character Interpretation: 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.
Award Snub: (Re: the 1995 film) Rumor has it Ian McKellen missed out on a Best Actor Oscar nomination by a single vote.
Complete Monster: Richard III is one of the most famous examples of a Historical Villain Upgrade in English drama. Richard 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 even 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.
Crowning Moment of Funny: 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.
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 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.
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!*.
Narm: Several serious scenes in the 1995 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.
Woolseyism: (Re: the 1995 film) McKellen modernized some of Shakespeare's dialogue, removing archaisms such as "thy" or "thou" and clarifying passages that relate to the Henry VI trilogy.