We discuss three topics related to the neutron star (NS) mass spectrum. At first we discuss the possibility to form low-mass ($ M stackrel{<}{sim} 1 M_{odot}$) and suggest this is possible only due to fragmentation of rapidly rotating proto-NSs. Such low-mass NSs should have very high spatial velocities which could allow identification. A critical assessment of this scenario is given. Secondly, we discuss mass growth due to accretion for NSs in close binary systems. With the help of numerical population synthesis calculations we derive the mass spectrum of massive ($M > 1.8 M_{odot}$) NSs. Finally, we discuss the role of the mass spectrum in population studies of young cooling NSs. We formulate a kind of {it mass constraint} which can be helpful, in our opinion, in discussing different competive models of the thermal evolution of NSs.
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