What do we (number theorists) do with ourselves now that Fermat's last theorem (FLT) has fallen? \[ \ \] I'll discuss numerous generalizations of FLT -- for instance, for fixed integers $a,b,c \geq 2$ satisfying $1/a + 1/b + 1/c < 1$, Darmon and Granville proved the single generalized Fermat equation $x^a + y^b = z^c$ has only finitely many coprime integer solutions. Conjecturally something stronger is true: for $a,b,c \geq 3$ there are no non-trivial solutions. More generally, I'll discuss my subfield "arithmetic geometry", and in particular the geometric intuitions that underlie the conjectural framework of modern number theory.