  This paper delves into the origins and specificity of the unique stable spin
textures (360{\deg} closed loop domain walls and skyrmions) observed in
exchange-biased systems, with either in-plane or out-of-plane magnetic
anisotropy. In the case of skyrmions, which are nanometer-sized bubbles
consisting of closed-loop 180{\deg} walls in perpendicularly-magnetized media,
the stability of these spin textures arises from the existence of
Dzyaloshinskii-Moriya Interactions (DMI). These interactions induce chirality
of the domain walls, yielding to some extent a so-called topological
protection. More complex structures such as skyrmoniums have been observed,
consisting of closed loop 360{\deg} walls. Recently, skyrmions formed in the
absence of an applied external magnetic field have been stabilized in exchange
biased out-of-plane magnetized systems. About two decades ago, another type of
stable spin-textures were observed in exchange biased systems, with in-plane
magnetization, in particular in the pinned reference layer of spin-valves.
These textures consist of 360{\deg}-domain-wall rings, the stability of which
arises from the easy-plane shape anisotropy of these layers. In this paper, we
compare these spin-textures and highlight the similarities and differences in
their formation, structure and origin of their stability.
