Complete intersections in projective space are, behind hypersurfaces, perhaps the easiest varieties to study, but even in simple cases can have interesting geometry. In the simplest non-trivial case of a complete intersection we can intersect two smooth quadric hypersurfaces, and even in this case the resulting variety X turns out to be worthy of study. I will discuss some of the interesting aspects, including the rationality of X and how one can associate to X a hyperelliptic curve (under a condition on the parity of dim X).