The Einstein equations admit an initial value formulation. A three-dimensional oriented manifold with a Riemannian metric g and a symmetric tensor K of rank 2 forms the corresponding vacuum initial data. The Gauss-Codazzi equations provide constraints on this initial data. The main aim of the booklet is to study properties of solutions of the constraint equations. J. Corvino and R. Schoen have recently developed a new method to study asymptotics of the vacuum constraint equations. The book describes a generalization of the method to a large class of weighted function spaces and, in particular, applications to various perturbation, gluing and extension results (including a proof of existence of initial data, which is exactly Kerrian outside of a compact set or a construction of large classes of initial data with controlled asymptotic behaviour).