  The determination of the absolute mass scale of neutrinos is one of the most
important challenges in Particle Physics. The shape of the endpoint region of
$\beta$-decay and electron capture (EC) spectra depends on the phase space
factor, which, in turn, is function of the neutrino mass eigenstates. High
energy resolution and high statistics measurements of $\beta$- and EC spectra
are therefore considered a model-independent way for the determination of the
neutrino mass scale. Since almost four decades, low temperature
microcalorimeters are used for the measurement of low energy $\beta$- and EC
spectra. The first efforts were focused on the development of large arrays for
the measurement of the $^{187}$Re $\beta$-spectrum. In the last ten years, the
attention moved to EC of $^{163}$Ho. This choice was mainly motivated by the
very good performance which could be achieved with low temperature
microcalorimeters enclosing $^{163}$Ho with respect to microcalorimeters with
absorber containing $^{187}$Re. The development of low temperature
microcalorimeters for the measurement of the finite neutrino mass is discussed
and, in particular, the reasons for moving from $^{187}$Re to $^{163}$Ho. The
possibility to reach sub-eV sensitivity on the effective electron neutrino mass
with $^{163}$Ho, thanks to the multiplexing of large microcalorimeter arrays is
demonstrated. In conclusion, an overview on other nuclides which have been
proposed as good candidates, motivated by the excellent performance of low
temperature microcalorimeters is presented.
