Magnetic reconnection provides an efficient conversion of the so-calle
d 'free magnetic energy' to kinetic and thermal energies of cosmic pla
smas, hard electromagnetic radiation, and accelerated particles. This
phenomenon was found in laboratory and space, but it is especially wel
l studied in the solar atmosphere where it manifests itself as flares
and flare-like events. We review the works devoted to the tearing inst
ability - the inalienable part of the reconnection process - in curren
t sheets which have, inside of them, a transverse (perpendicular to th
e sheet 'plain') component of the magnetic field and a longitudinal (p
arallel to the electric current) component of the field. Such 'non-neu
tral' current sheets are well known as the energy sources for flare-li
ke processes in the solar corona. In particular, quasi-steady high-tem
perature turbulent current sheets are the energy sources during the 'm
ain' or 'hot' phase of solar flares. These sheets are stabilized with
respect to the collisionless tearing instability by a small transverse
component of magnetic field, normally existing in the reconnecting an
d reconnected magnetic fluxes. The collision tearing mode plays, howev
er, an important and perhaps dominant role for non-neutral current she
ets in solar flares. In the MHD approximation, the theory shows that t
he tearing instability can be completely stabilized by the transverse
field B(n) if its value satisfies the condition B(n)/B much greater th
an S-3/4. B is the reconnecting component of the magnetic field just n
ear the current sheet, S is the magnetic Reynolds number for the sheet
. In this case, stable current sheets become sources of temporal spati
al oscillations and usual MHD waves. The application of the theory to
the solar atmosphere shows that the effect of the transverse field exp
lains high stability of high-temperature turbulent current sheets in t
he solar corona. The stable current sheets can be sources of radiation
in the radio band. If the sheet is destabilized (at B(n)/B much less
than S-3/4), the compressibility of plasma leads to the arizing of the
tearing instability in a long wave region, in which for an incompress
ible plasma the instability is absent. When a longitudinal magnetic fi
eld exists in the current sheet, the compressibility-induces instabili
ty can be dumped by the longitudinal field. These effects are signific
ant in destabilization of reconnecting current sheets in solar flares:
in particular, the instability with respect to disturbances comparabl
e with the width of the sheet is determined by the effect of compressi
bility.