  This paper has two main goals. First, we are concerned with the
classification of self-adjoint extensions of the Laplacian
$-\Delta\big|_{C^\infty_0(\Omega)}$ in $L^2(\Omega; d^n x)$. Here, the domain
$\Omega$ belongs to a subclass of bounded Lipschitz domains (which we term
quasi-convex domains), which contain all convex domains, as well as all domains
of class $C^{1,r}$, for $r\in(1/2,1)$. Second, we establish Krein-type formulas
for the resolvents of the various self-adjoint extensions of the Laplacian in
quasi-convex domains and study the properties of the corresponding
Weyl--Titchmarsh operators (or energy-dependent Dirichlet-to-Neumann maps).
  One significant technical innovation in this paper is an extension of the
classical boundary trace theory for functions in spaces which lack Sobolev
regularity in a traditional sense, but are suitably adapted to the Laplacian.
