> Anyone have a good explanation for why elliptic curves have a 'natural' group law? [...] As far as I've seen, the group law is what makes elliptic curves special. Are they the _only_ flavour of curve that has a nice geometric group law?

I asked the same question to a professor who works in topics related to algebraic geometry. His answer was very simple: it's because elliptic curves form Abelian varieties

> https://en.wikipedia.org/wiki/Abelian_variety

i.e. a projective variety that is also an algebraic group

> https://en.wikipedia.org/wiki/Algebraic_group

Being an algebraic group means that the group law on the variety can be defined by regular functions.

Basically, he told to read good textbooks about abelian varieties if one is interested in this topic.

> Are they the _only_ flavour of curve that has a nice geometric group law?

The Jacobian of a hyperelliptic curve (which generalize elliptic curves) also forms an abelian variety. Its use in cryptography is named "hyperelliptic curve cryptography":

> https://en.wikipedia.org/wiki/Hyperelliptic_curve_cryptograp...