The structure of rational solutions to a polynomial equation depends on the structure of the corresponding algebraic variety. In case of a curve of genus zero, the problem of finding all solutions can be completely solved using the Hasse-Minkowski principle. In case of genus one, the obstruction to the Hasse-Minkowski principle is conjectured to be finite; and the set of rational points is a finitely generated group by the Mordell-Weil theorem if it is not empty. In case of genus two or bigger, the set of solutions is finite by Faltings' theorem.
A major unsolved problem today is the effectivity of solutions for curves of genus one or bigger. For elliptic curves, one has the Birch and Swinnerton-Dyer (BSD) conjecture which relates the Mordell-Weil group and the central values of L-series arising from counting rational points over finite fields. For curves of genus two or bigger, one has the ABC conjecture and its refinements providing some effective bounds for curves. In the function field case, these conjectures are consequences of Bogomolov-Miyaoka-Yau.