A new phenomenon, th, fermion condensation in Fermi systems with stron
gly repulsive interaction, is analyzed. This phenomenon is shown to oc
cur when the necessary stability condition of the normal ground-state
quasiparticle distribution n(F)(p) = theta(p - p(F)) is violated and t
he rearrangement of this distribution occurs. The presence of the ferm
ion condensate means that a fraction of quasiparticles with different
momenta p has the same single-particle energy epsilon(p) which is equa
l to the chemical potential mu that results in the essential enhanceme
nt of the density of states similar to that of a Bose liquid below the
critical temperature T(c). The microscopic theory of the evaluation o
f the critical point parameters is outlined. It is found that this pha
se transition could occur in a low-density electron gas with r(s) = 10
-15. Various properties of the systems with the fermion condensate are
investigated within simple solvable models. The superfluid correlatio
ns in systems with the fermion condensate are also considered. The gap
DELTA in the single-particle excitation spectra of such systems is sh
own to have not the usual exponential BCS-smallness.