  An integral domain $D$ is called an irreducible-divisor-finite domain
(IDF-domain) if every nonzero element of $D$ has finitely many irreducible
divisors up to associates. The study of IDF-domains dates back to the
seventies. In this paper, we investigate various aspects of the IDF property.
In 2009, P.~Malcolmson and F. Okoh proved that the IDF property does not ascend
from integral domains to their corresponding polynomial rings, answering a
question posed by D. D. Anderson, D. F. Anderson, and the second author two
decades before. Here we prove that the IDF property ascends in the class of
PSP-domains, generalizing the known result (also by Malcolmson and Okoh) that
the IDF property ascends in the class of GCD-domains. We put special emphasis
on IDF-domains where every nonunit is divisible by an irreducible, which we
call TIDF-domains, and we also consider PIDF-domains, which form a special
class of IDF-domains introduced by Malcolmson and Okoh in 2006. We investigate
both the TIDF and the PIDF properties under taking polynomial rings and
localizations. We also delve into their behavior under monoid domain and $D+M$
constructions
