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A conic bundlef:X→Sis a flat morphism of smooth varieties whose fibers are planeconics. In this talk, I will first discuss application of Sarkisov program to the rationality problemof algebraic varieties having conic bundle structures. Then I concentrate on some special, so-calledtetragonal types of conic bundles, which lie on the “boundary” between birationally rigid and non-rigid ones and are especially interesting for this reason. The second part of the talk is based on thejoint work in progress with V. Shokurov.