Napoleon Solo's trick of removing a tablecloth from a table while leaving all the objects undisturbed was not a visual effect. Henry Cavill actually performed it himself, having been trained in the trick by British variety star Mat Ricardo.
Henry Cavill and Armie Hammer often performed their own stunts. Hammer was more enthusiastic about his stunts, to the point that his stunt double joked that he hardly had a chance to do anything because he was out there doing it all by himself.
Henry Cavill admitted that he was very relieved to not have a shirtless scene in this movie after a string of movies where he had to be in top shape. Cavill said that he was glad to focus on his performance for once and not his physique, which he said took an enormous amount of work to present on screen.
Armie Hammer watched The Man from U.N.C.L.E. (1964) before starting production to prepare for his role as Illya Kuryakin. He was the only member of the cast to do so.
The suits that Henry Cavill wears as Napoleon Solo were created by renowned British tailor Timothy Everest, who has also worked on the James Bond, and Mission: Impossible films. Solo's suits feature unique details such as covered buttons and squared-off front edges that were referenced from the late 1960s. During one of his costume fittings at Everest's shop, Henry Cavill accidentally broke the change room door off its hinges. The staff kept it as a memento of the time that "Superman" broke their door.
David Beckham: as the projectionist at the beginning of the movie, when Illya is being briefed about Solo.