  We discuss the properties of topologically nontrivial superconducting phases
and the conditions for their realization in condensed matter, and the
principles for identifying Majorana bound states (MBSs). Along with the
well-known Kitaev chain and superconducting nanowire (SW) models with
spin-orbit coupling in an external magnetic field, we discuss models of
quasi-2D materials in which MBSs are realized in the presence of noncollinear
spin ordering. For finite-length SWs, we demonstrate a cascade of quantum
transitions occurring with a change in the magnetic field, accompanied by a
change in the fermion parity of the ground state. The corresponding anomalous
behavior of the magnetocaloric effect can be used as a tool for identifying
MBSs. We devote considerable attention to the analysis of the transport
characteristics of devices that contain topologically nontrivial materials. The
results of studying the conductance of an Aharonov-Bohm ring whose arms are
connected by an SW are discussed in detail. An important feature of this device
is the appearance of Fano resonances in the dependence of conductance on the
magnetic field when the SW is in a topologically nontrivial phase. We establish
a relation between the characteristics of such resonances and the spatial
structure of the lowest-energy SW state. The conditions for the occurrence of
an MBS in the phase of the coexistence of chiral $d+id$ superconductivity and
120-degree spin ordering are determined in the framework of the $t-J-V$ model
on a triangular lattice. We take electron-electron interactions into account in
discussing the topological invariants of low-dimensional superconducting
materials with noncollinear spin ordering. The formation of Majorana modes in
regions with an odd value of a topological $\mathbb{Z}$ invariant is
demonstrated. The spatial structure of these excitations in the Hubbard fermion
ensemble is determined.
