V.Tarasov. B. and M.Shapiro conjecture and Bethe ansatz Abstract: The B. and M. Shapiro conjecture is the following statement: If the Wronskian of a set of polynomials with complex coefficients has only real roots, then the complex span of this set of polynomials has a basis consisting of polynomials with real coefficients. The conjecutre has several implications in enumerative real algebraic geometry. For the case of two polynomials, the conjecture was proved by A.Eremenko and A.Gabrielov in 2002. In general, the conjecture was proved recently by E.Mukhin, A.Varchenko and the speaker using the algebraic Bethe ansatz for the $gl_n$ Gaudin model.