  We study long time behavior of integrated trawl processes introduced by
Barndorff-Nielsen. The trawl processes form a class of stationary infinitely
divisible processes, described by an infinitely divisible random measure
(L\'evy base) and a family of shifts of a fixed set (trawl). We assume that the
L\'evy base is symmetric and homogeneous and that the trawl set is determined
by the trawl function that decays slowly. Depending on the geometry of the
trawl set and on the L\'evy measure corresponding to the L\'evy base we obtain
various types of limits in law of the normalized integrated trawl processes for
large times. The limit processes are always stable and self-similar with
stationary increments. In some cases they have independent increments - they
are stable L\'evy processes where the index of stability depends on the
parameters of the model. We show that stable limits with stability index
smaller than 2 may appear even in cases when the underlying L\'evy base has all
its moments finite. In other cases, the limit process has dependent increments
and it may be considered as a new extension of fractional Brownian motion to
the class of stable processes.
