How many eigenvalues of a random matrix are real ?

Let M_n be a random matrix of size n whose entries are iid real random variables with mean 0 and variance 1.
As M_n is non-hermitian, one expects that most of its eigenvalues are complex. However, as the eigenvalues
are roots of the characteristic polynomial, one also expects that few of them are real (for instance of n is odd, then one must have at least one real roots). The question is: How many ? More deeply, what forces
the eigenvalues to be real (rather than trivial algebraic reason such as above) ?

I am going to give a brief survey on this question and then present a recent result (obtained jointly with T. Tao), which gives a partial answer for this question, and discuss several open questions.