First, to all the Americans on here outraged about the film "endorsing" a criminal act - NEWS FLASH!! Just because the age of consent is 18 in your country, doesn't make it so everywhere else in the world! 18 is comparatively old by global standards. I believe the average age is 16 (as in my country) and in Italy where the film takes place it's actually 14. Whether you agree with it morally or not, no criminal act takes place - US law does not apply!
However, I agree with many reviewers that Armie Hammer is miscast in terms of his character's age. He is 31 and looks it, whereas Oliver is meant to be 24. His age is never referred to in the film, so viewers who haven't read the book are left with the impression of a 30-ish man engaging in a relationship with a 17-year-old, which Timothee (20 at the time) does convincingly portray. I can fully understand how that would ring alarm bells for many. If they'd had an actor who actually looked 24 I think that would have created a very different dynamic. Even so, the relationship is hardly of a paedophilic nature. And I don't think most people would regard 17-year-olds as "children" physically and emotionally, even if they may be legally.
Timothee Chalamet gives an incredible performance fully deserving of all the accolades, but I found Armie's inconsistent at best. He does get better as the film goes along. Maybe it's just that I didn't find his character particularly appealing... to me he comes across as rather abrupt and arrogant, and weirdly passive-aggressive and patronising towards Elio for much of the time. The physical scenes between Oliver and Elio were excruciatingly awkward to watch and didn't ring true to me - but I'm a straight female so what would I know? I haven't had that reaction with the (admittedly few) other gay films I've seen though. Maybe it was supposed to reflect real life, or maybe I am just too accustomed to male-female depictions. Ultimately the relationship seemed to consist of Elio being infatuated and Oliver enjoying being the object of that infatuation, rather than a true love story.
Personally the relationship I found most touching in the film was that between Elio and his parents. Although they aren't given much to work with, both the actors playing his parents do beautiful, nuanced work... and the much-lauded scene near the end with his father was stunning and truly moving. It was refreshing to see a such a loving and accepting child-parent relationship, done in a natural and non-affected way. It rang very true to my own experience as an only child and it was nice to see this portrayed on film for once, instead of the tension-filled relationships American films usually depict.
Overall I thought the film was absorbing and well-done, but neither the masterpiece or great love story it's being hyped as.