  The paramagnetic Meissner effect (PME) is observed in small superconducting
samples, and a number of controversial explanations of this effect are
proposed, but there is as yet no clear understanding of its nature. In the
present paper PME is considered on the base of the Ginzburg-Landau theory (GL).
The one-dimensional solutions are obtained in a model case of a long
superconducting cylinder for different cylinder radii R, the GL-parameters
\kappa and vorticities m. Acording to GL-theory, PME is caused by the presence
of vortices inside the sample. The superconducting current flows around the
vortex to screeen the vortex own field from the bulk of the sample. Another
current flows at the boundary to screen the external field H from entering the
sample. These screening currents flow in opposite directions and contribute
with opposite signs to the total magnetic moment (or magnetization) of the
sample. Depending on H, the total magnetization M may be either negative
(diamagnetism), or positive (paramagnetism). A very complicated saw-like
dependence M(H) (and other characteristics), which are obtained on the base of
self-consistent solutions of the GL-equations, are discussed.
