A classical approach to the problem of effectively parametrizing the space of solutions to the vacuum Einstein constraint equations is the conformal method. We will focus on the constant mean curvature case, which reduces to the analysis of the Lichnerowicz equation to solve the Hamiltonian constraint, and which brings the study of the scalar curvature operator to the fore. We begin the chapter with elements of the theory of elliptic partial differential equations, and then develop the conformal method and present in detail the closed constant mean curvature case. In the last section we discuss more general scalar curvature deformation and an obstruction to positive scalar curvature due to Schoen and Yau.