  The review is aimed at highlighting the aspects of topological
superconductivity in the absence of spin-orbit interaction in two-dimensional
systems with long-range non-collinear spin ordering or magnetic skyrmions.
Another purpose is to give a brief introduction to the new concept of
topological superconductivity, i.e. higher-order topology in two-dimensional
systems including spin-orbit coupled structures. The formation of Majorana
modes due to magnetic textures is discussed. The role of effective triplet
pairings and odd fermion parity of the ground state wave function in different
systems is emphasized. We describe the peculiarities of the magnetic skyrmions,
leading to the formation of the Majorana modes and defects on which the modes
are localized. The problem of braiding in the two-dimensional systems,
especially in higher-order topological superconductors, is considered.
