TITLE: Why certain diophantine equations are interesting (and others are not) ?
QUESTION [59 upvotes]: It is quite clear why certain differential equations, among the jungle of possible diff equations that is possible to conceive, are studied: some come from physical problems, or from "spontaneous" mathematical generalizations thereof, others come from geometry in a variety of ways. 
For diophantine equations there seem not to be such a direct link to other areas.
I would like to roughly understand why the attention of number theorists concentrates on some kinds of diophantine equations and not on others. 
Why an equation such as 
$x^2-ny^2=1$
or
$x^3+y^3=z^3$
is (or have been) considered worth studying, and not, say, any other random variant such as (if that specific example is not enough nontrivial for you or if it actually happens to have been studied, feel free to substitute it with your favourite "random" diophantine equation):
$x^3+y^5=z^2$  ?  So:

Are there any reasons why certain diophantine equations are worth attention besides the mere approachability (i.e. being neither trivial nor hopelessly difficult to analyze)?

REPLY [6 votes]: 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.