Authors’ summary: The quadratic matrix equation \(AX^2 + BX + C = 0\) in \(n \times n\) matrices arises in applications and is of intrinsic interest as one of the simplest nonlinear matrix equations. We give a complete characterization of solution in terms of the generalized Schur decomposition and describe and compare various numerical solution techniques. In particular, we give a thorough treatment of functional iteration methods based on Bernoulli’s method. Other methods considered include Newton’s method with exact line searches, symbolic solution and continued fractions. We show that functional iteration applied to the quadratic matrix equation can provide an efficient way to solve the associated quadratic eigenvalue problem \((\lambda^2 A + \lambda B + C) x = 0\).