This talk will be about rational points on algebraic varieties
which arise as the common zero locus of two quadratic forms.
The main problem is to understand the asymptotics of the
counting function for the number of rational points upto a
specific height. In general the problem is open, and gets
harder as the number of variables decreases. I will report
on some recent works in this direction, a part of which is
joint with Henryk Iwaniec, and a part with Tim Browning.
Our approach is analytic and is based
on the circle method.