Bill Murray stars as Don Johnston, and on the off chance that this is too subtle a character name for you, he's watching an old movie about Don Juan when we meet him, and several other characters point out that he's a regular Don Juan himself. His most recent conquest (Julie Delpy) has just left him when he receives an anonymous letter from an old girlfriend, telling him that he has a 19-year-old son.
Don isn't particularly motivated to do anything about the letter -- he doesn't seem motivated to do much of anything about anything -- but his neighbor Winston (Jeffrey Wright, giving the movie's best performance) insists that Don go on a road trip to visit all of the potential mothers/ex-girlfriends and try to solve the mystery. So off goes Don to visit four exes (played by Sharon Stone, Frances Conroy, Jessica Lange, and Tilda Swinton).
If you were annoyed by BIll Murray in Lost in Translation, or if you didn't like that movie much, you'll hate Broken Flowers; Translation was a roller-coaster ride of action by comparison. Very, very, very little happens in Broken Flowers, and it happens extremely slowly. Murray, whose recent films have pushed the limits of catatonia as an acting style, goes well beyond those limits here; Don has no apparent personality, and we have no clue what might have attracted these women to him 20 years ago.
Or him to them, for that matter; they are almost as devoid of personality as he is. His reunions are (with the exception of Stone's character, the most interesting of the women) awkward and uncomfortable, and it's difficult to imagine any of these couples as having been happy together for even an instant.
The flatness and anonymity of the characters in the movie extends to its settings; we have only the vaguest clues of landscape to tell us where any of these people live (I'd bet on upstate New York or northern New England for Swinton, and we know that Don's local airport is served by Northwest, but that's the extent of our knowledge).
Were it not for Wright's lively, funny performance as Winston, there'd be nothing to recommend the movie, and even Wright's fine work isn't enough for me to suggest that you should buy a ticket.