One of Manjul Bhargava's most surprising results in arithmetic geometry is his proof that most hyperelliptic curves of a fixed genus (at least 2) which are defined over the rational numbers have no rational points.

I will review the arguments in his proof, which involve pencils of quadrics and homogenous spaces for the Jacobian. I will also discuss some joint work with Bhargava and Wang, where we show that a positive proportion of hyperelliptic curves of a fixed genus which are defined over the rational numbers have no points over any extension of odd degree.