Rationality problem for conic bundles

A conic bundle is a flat morphism $f: X\to Z$ of smooth algebraic varieties whose fibers are plane conics. I will discuss the problem of rationality of algebraic varieties having conic bundle structures. First, I recall almost classical results on birational properties of surface conic bundles over non-closed fields. Then I concentrate on the three-dimensional case. The main focus will be on the conjectural criterion of rationality.