Deciding whether a given algebraic variety is rational (birational to projective space) is an important problem in algebraic geometry. Over the field of real numbers, this problem is particularly interesting for varieties that are known to be rational over the complex numbers, as it then has an arithmetic flavour. In this talk, I will review the techniques that are available, and focus on concrete examples of real conic bundles for which I will provide new positive and negative results. This is joint work with Alena Pirutka. -- 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.