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Shell-model studies on the weak $\beta$-decay in nuclei relevant to astrophysical processes are carried out. The $\beta$-decay rates, as well as electron-capture rates in the $sd$- $pf$shell induced by Gamow–Teller (GT) transition, are evaluated in astrophysical environments. The weak rates for the Urca pair of nuclei with $A$= 31 in the island of inversion, which are important for the nuclear Urca processes in neutron star crusts, are investigated by shell-model calculations in the $sd$– $pf$shell. The GT strength is evaluated in the $sd$– $pf$shell for selected $\beta$-decays in the $sd$-shell nuclei, and the effects of the expansion of the configuration space on the quenching of the axial–vector coupling are examined. $\beta$-decay rates induced by first-forbidden (FF) transitions are studied by the Behrens–Bühring (BB) method for the isotones with $N$= 126 and compared with the Walecka method. The important role of the electron distortions in the $\beta$-decays of²⁰⁶Hg and²⁰⁷Tl is pointed out.