Computing rational solutions to $A{x}^p+B{y}^p+C{z}^p=0$

In this survey, we studied the possibility of finding rational solutions to the equation $A{x}^p+B{y}^p+C{z}^p=0$ via its attached hyperelliptic curve $Y^2=X^p+A^2(BC{)}^{p-1}\mathrm{/}4$ and its rational points computed using computational tools.