A Survey on the Ternary Purely Exponential Diophantine Equation $a^x+b^y=c^z$

Let $a$, $b$, $c$ be fixed coprime positive integers with $\min \{a,b,c\}>1$. In this survey, we consider some unsolved problems and related works concerning the positive integer solutions $(x,y,z)$ of the ternary purely exponential diophantine equation $a^x+b^y=c^z$.