  We present three families of pairs of geometrically non-isomorphic curves
whose Jacobians are isomorphic to one another as unpolarized abelian varieties.
Each family is parametrized by an open subset of P^1. The first family consists
of pairs of genus-2 curves whose equations are given by simple expressions in
the parameter; the curves in this family have reducible Jacobians. The second
family also consists of pairs of genus-2 curves, but generically the curves in
this family have absolutely simple Jacobians. The third family consists of
pairs of genus-3 curves, one member of each pair being a hyperelliptic curve
and the other a plane quartic. Examples from these families show that in
general it is impossible to tell from the Jacobian of a genus-2 curve over Q
whether or not the curve has rational points -- or indeed whether or not it has
real points. Our constructions depend on earlier joint work with Franck
Leprevost and Bjorn Poonen, and on Peter Bending's explicit description of the
curves of genus 2 whose Jacobians have real multiplication by Z[\sqrt{2}].
