  Nuclear weak rates in stellar environments are obtained by shell-model
calculations including Gamow-Teller (GT) and spin-dipole transitions, and
applied to nuclear weak processes in stars. The important roles of accurate
weak rates for the study of astrophysical processes are pointed out. The weak
rates in $sd$-shell are used to study the evolution of ONeMg cores in stars
with 8-10 M$_{\odot}$. Cooling of the core by nuclear Urca processes, and the
heating by double e-captures on $^{20}$Ne are studied. Especially, the
e-capture rates for a second-forbidden transition in $^{20}$Ne are evaluated
with the multipole expansion method of Walecka and
Behrens-B$\ddot{\mbox{u}}$hring, and the final fate of the cores, core-collapse
or thermonuclear explosion, are discussed. The weak rates in $pf$-shell are
applied to nucleosynthesis of iron-group elements in Type Ia supernovae. The
over-production problem of neutron-rich iron isotopes compared with the solar
abundances is now reduced to be within a factor of two. The weak rates for
nuclear Urca pair with $A$=31 in the island of inversion are evaluated with the
effective interaction obtained by the extended Kuo-Krenciglowa method. The
transition strengths and e-capture rates in $^{78}$Ni, important for
core-collapse processes, are evaluated with the $pf$-$sdg$ shell, and compared
with those obtained by the random-phase-approximation and an effective rate
formula. $\beta$-decay rates of $N$ =126 isotones are evaluated with both the
GT and first-forbidden transitions. The half-lives are found to be shorter than
those obtained by standard models. Neutrino-nucleus reaction cross sections on
$^{13}$C, $^{16}$O and $^{40}$Ar are obtained with new shell-model
Hamiltonians. Implications on nucleosynthesis, neutrino detection, neutrino
oscillations and neutrino mass hierarchy are discussed.
