Neutron star matter with densities exceeding that of the normal nuclear matter (no ≈ 0.17/fm3) is discussed in light of the recent developments in understanding of the structure of the nucleon in terms of its basic constituents : quarks and gluons. Two different scenarios of phase transition-pion condensation followed by quark matter or percolation followed by quark plasma-are possible depending upon different confinement sizes : the "Little Bag" or the "Big Bag" (M.I.T. bag). Consequences on neutron star properties are briefly discussed.