  The work of Adler provides necessary and sufficient conditions for the
Wronskian of a given sequence of eigenfunctions of Schr\"odinger's equation to
have constant sign in its domain of definition. We extend this result by giving
explicit formulas for the number of real zeros of the Wronskian of an arbitrary
sequence of eigenfunctions. Our results apply in particular to Wronskians of
classical orthogonal polynomials, thus generalizing classical results by Karlin
and Szeg\H{o}. Our formulas hold under very mild conditions that are believed
to hold for generic values of the parameters. In the Hermite case, our results
allow to prove some conjectures recently formulated by Felder et al.
