Many physical models admit an initial value formulation. In this chapter we discuss an initial value formulation for the vacuum Einstein equation. A vacuum initial data set will be given geometrically as a manifold endowed with Riemannian metric and a symmetric two-tensor. That these give the first and second fundamental forms of an embedding into a Lorentzian manifold satisfying the vacuum Einstein equation imposes, via the Gauss and Codazzi equations, constraints on the initial data. These conditions, which govern the space of allowable initial data sets for the vacuum Einstein equation, comprise the Einstein constraint equations, the study of solutions to which form an interesting and rich subject for geometric analysis.