  This is a survey paper of author's results on cobordism groups and semigroups
of fold maps and simple fold maps. The results include: establishing a relation
between fold maps and immersions through geometrical invariants of cobordism
classes of fold maps and simple fold maps in terms of immersions with
prescribed normal bundles, detecting stable homotopy groups of spheres as
direct summands of the cobordism semigroups of fold maps, Pontryagin-Thom type
construction for -1 codimensional fold maps and estimations about the cobordism
classes of manifolds which have fold maps into stably parallelizable manifolds.
In the last section some of these results are extended and we show that our
invariants also detect stable homotopy groups of the classifying spaces $BO(k)$
as direct summands of the cobordism semigroups of fold maps.
