In the final lecture, we will focus on the uniqueness question for Einstein metrics. That is, what can we say about the space of all Einstein metrics on a given  4-manifolds, modulo diffeomorphisms and rescalings? We will carefully discuss some key examples where this Einstein  moduli space can be shown to be connected. We will then discuss some partial results concerning other important 4-manifolds.  However, we will also see that
there do exist 4-manifolds for which the Einstein moduli space fails to be connected.  The lecture will then conclude with a discussion of open problems.