British Film Institute archivist David Meeker compiled this excellent, quirky list of mostly standard and some slightly off-canon international films, which was published in 1998 in Sight and Sound. It wisely eschewed recency bias by ending the list at 1981. As with any list there are blind spots but 360 (actually 365) films is a small number and any list of worthy movies can always be nitpicked for one reason or agenda or another. As of May 20, 2018, I have seen all but 21 of these, or 344 out of the 365 total, or 94 percent of the list. I'm posting it here because I'm not aware that anyone else has done so on Letterboxd.
There are 365 films on the list because the idea was to have a one-year repertory of one great film for the BFI to screen each day for one year. Not sure why it was named "360 Film Classics" instead of "365 Classics" unless series like the Apu and Human Condition trilogies, etc., are supposed to count as one.