  We lay the foundations of convex hypersurface theory (CHT) in contact
topology, extending the work of Giroux in dimension three. Specifically, we
prove that any closed hypersurface in a contact manifold can be
$C^0$-approximated by a convex one. We also prove that a $C^0$-generic family
of mutually disjoint closed hypersurfaces parametrized by $t \in [0,1]$ is
convex except at finitely many times $t_1, \dots, t_N$, and that crossing each
$t_i$ corresponds to a bypass attachment. As applications of CHT, we prove the
existence of compatible (relative) open book decompositions for contact
manifolds and an existence h-principle for codimension 2 contact submanifolds.
