The movie is set in the 1960s. Henry Cavill plays Napoleon Solo, who we first see visiting East Berlin to extract Gaby Teller (Alicia Vikander). They're pursued by Soviet agent Illya Kuryakin (Armie Hammer), who comes close to ruining Solo's rather ornate but effective escape plan. The very next day, Napoleon and Gaby are told they'll have to work with Illya because Gaby's father appears to be building atomic bombs for the Nazi sympathizers who are holding him captive. So none of them like each other (a novel idea).
The movie is an amazingly traditional action-spy-comedy in the mold of almost every one that's come before. And that strikes me as extremely odd, given that it was directed by Guy Ritchie - not a man known for following conventions. The movie does manage some rather lovely set pieces: unfortunately, Cavill and Hammer are poor enough actors that neither manages to build a memorable character with the lines they're given. Although this isn't entirely their fault as the script is no masterpiece, with "character" being replaced with "a collection of tics" and Vikander - a better actor than either - struggling with the same problem.