A variety is called rational if it contains a Zariski open subset isomorphic to an open subset of projective space. Determining whether a given smooth projective variety is not rational is often a complex problem. In this seminar, we present a criterion for irrationality for conic bundles over the complex numbers, based on their behavior in characteristic 2. By exploiting the pathological properties that such bundles exhibit in this characteristic, we apply the criterion to several examples. This is joint work with Giuseppe Ancona. -- This seminar is part of the activities of the Dipartimento di Eccellenza CUP B83C23001390001 and it is funded by the European Union – Next Generation EU.