  We review in detail the construction of {\em all} stable static fermion bags
in the 1+1 dimensional Gross-Neveu model with $N$ flavors of Dirac fermions, in
the large $N$ limit. In addition to the well known kink and topologically
trivial solitons (which correspond, respectively, to the spinor and
antisymmetric tensor representations of O(2N)), there are also threshold bound
states of a kink and a topologically trivial soliton: the heavier topological
solitons (HTS). The mass of any of these newly discovered HTS's is the sum of
masses of its solitonic constituents, and it corresponds to the tensor product
of their O(2N) representations. Thus, it is marginally stable (at least in the
large $N$ limit). Furthermore, its mass is independent of the distance between
the centers of its constituents, which serves as a flat collective coordinate,
or a modulus. There are no additional stable static solitons in the Gross-Neveu
model. We provide detailed derivation of the profiles, masses and fermion
number contents of these static solitons. For pedagogical clarity, and in order
for this paper to be self-contained, we also included detailed appendices on
supersymmetric quantum mechanics and on reflectionless potentials in one
spatial dimension, which are intimately related with the theory of static
fermion bags. In particular, we present a novel simple explicit formula for the
diagonal resolvent of a reflectionless Schr\"odinger operator with an arbitrary
number of bound states. In additional appendices we summarize the relevant
group representation theoretic facts, and also provide a simple calculation of
the mass of the kinks.
