Scalar Curvature and the Einstein Constraint Equations

We survey some results on scalar curvature and properties of solutions to the Einstein constraint equations. Topics include an extended discussion of asymptotically flat solutions to the constraint equations, including recent results on the geometry of the center of mass of such solutions. We also review methods to construct solutions to the constraint equations, including the conformal method, as well as gluing techniques for which it is important to understand both conformal and non-conformal deformations of the scalar curvature.