As has been well told by others, there are many interesting classes of diopantine equations (norm equations, elliptic curves and abelian varieties, curves of genus > 1, S-unit equations, varieties where the rational points are always Zariski dense or are always finite, failure of the Hasse principal (or not), Lang's conjecture, etc.). However, it is a theorem of Wiles that are no more specific diophantine equations of interest. Any particular problem, say, the existence of infinitely many integral solutions to $x^3+y^3+z^3 = 3$, will only be interesting to the extent that the solution sheds light on the general arithmetic properties of surfaces. Fermat was special, for a combination of historic and aesthetic reasons.

As has been well told by others, there are many interesting classes of diopantine equations (norm equations, elliptic curves and abelian varieties, curves of genus > 1, S-unit equations, varieties where the rational points are always Zariski dense or are always finite, failure of the Hasse principle (or not), Lang's conjecture, etc.). However, it is a theorem of Wiles that are no more specific diophantine equations of interest. Any particular problem, say, the existence of infinitely many integral solutions to $x^3+y^3+z^3 = 3$, will only be interesting to the extent that the solution sheds light on the general arithmetic properties of surfaces. Fermat was special, for a combination of historic and aesthetic reasons.