Let $F$ be an irreducible homogeneous polynomial in two variables with integer coefficients and with degree at least 3. Consider the equation $F(x,y) = h$ for some fixed non-zero integer $h$. In a pioneering work in 1909, Thue proved that this equation has only finitely many integral solutions. Much effort has been made to obtain upper bounds for the number of solutions of these equations which are independent of the size of the coefficients of $F$. In this talk, we discuss the upper bounds predicted by some of the central conjectures of this area and present some partial contributions in that direction.
