  We study the boundary rigidity problem for compact Riemannian manifolds with
boundary $(M,g)$: is the Riemannian metric $g$ uniquely determined, up to an
action of diffeomorphism fixing the boundary, by the distance function
$\rho_g(x,y)$ known for all boundary points $x$ and $y$? We prove in this paper
global uniqueness and stability for the boundary rigidity problem for generic
simple metrics. More specifically, we show that there exists a generic set
$\mathcal{G}$ of simple Riemannian metrics and an open dense set
$\mathcal{U}\subset \mathcal{G}\times\mathcal{G}$, such that any two Riemannian
metrics in $\mathcal{U}$ having the same distance function, must be isometric.
We also prove H\"older type stability estimates for this problem for metrics
which are close to a given one in $\mathcal{G}$.
