  We consider the critical points of Steklov eigenfunctions on a compact,
smooth $n$-dimensional Riemannian manifold $M$ with boundary $\partial M$. For
generic metrics on $M$ we establish an identity which relates the sum of the
indexes of a Steklov eigenfunction, the sum of the indexes of its restriction
to $\partial M$, and the Euler characteristic of $M$. In dimension $2$ this
identity gives a precise count of the interior critical points of a Steklov
eigenfunction in terms of the Euler characteristic of $M$ and of the number of
sign changes of $u$ on $\partial M$. In the case of the second Steklov
eigenfunction on a genus $0$ surface, the identity holds for any metric. As a
by-product of the main result, we show that for generic metrics on $M$ Steklov
eigenfunctions are Morse functions in $M$.
