The alternating direction method of multipliers (ADMM) is a versatile approach for solving multi-block separable optimization problems. This chapter focuses on ADMM for two-block convex separable optimization problems with equality constraints. We establish the convergence of ADMM and its proximal variant, deriving both ergodic and non-ergodic convergence rates. Additionally, we analyze the linear convergence of proximal ADMM and discuss an accelerated ADMM. For a broader discussion on the convergence of ADMM for multi-block separable optimization problems, we refer the reader to the survey paper.