Most of the world puts on a talis at Bar Mitzvah or earlier. In Eastern Europe, the minhag was to put one on only at marriage. This of course refers only to the larger tallis that is worn as a cloak. The smaller tallis which is worn as a shirt is put on in early childhood (when toilet trained, at age 3, or thereabouts).
One theory is this has to do with historic poverty. The idea be that until marriage you haven’t become a grown up and therefore shouldn’t go around dressing like one.