  I provide methods of constructing elliptic and hyperelliptic curves over
global fields with interesting rational points over the given fields or over
large field extensions. I also provide a elliptic curves defined over any given
number field equipped with a rational point, (resp. with two rational points)
of infinite order over the given number field, and elliptic curves over the
rationals with two rational points over `simplest cubic fields.' I also provide
hyperelliptic curves of genus exceeding any given number over any given number
fields with points (over the given number field) which span a subgroup of rank
at least $g$ in the group of rational points of the Jacobian of this curve. I
also provide a method of constructing hyperelliptic curves over rational
function fields with rational points defined over field extensions with large
finite simple Galois groups, such as the Mathieu group $M_{24}$.
