  We prove the $l^2$ Decoupling Conjecture for compact hypersurfaces with
positive definite second fundamental form and also for the cone. This has a
wide range of important consequences. One of them is the validity of the
Discrete Restriction Conjecture, which (up to $N^\epsilon$ losses) implies the
full range of expected $L^p_{x,t}$ Strichartz estimates for both classical and
irrational tori. Another one is an improvement in the range for the discrete
restriction theory for lattice points on the sphere. Various applications in
Additive Combinatorics, Incidence Geometry and Number Theory are also
discussed. Our argument relies on the interplay between linear and multilinear
restriction theory.
