AbstractWe discuss a number of properties of CMV matrices, by which we mean the class of unitary matrices studied recently by Cantero, Moral, and Velázquez. We argue that they play an equivalent role among unitary matrices to that of Jacobi matrices among all Hermitian matrices. In particular, we describe the analogues of well‐known properties of Jacobi matrices: foliation by co‐adjoint orbits, a natural symplectic structure, algorithmic reduction to this shape, Lax representation for an integrable lattice system (Ablowitz‐Ladik), and the relation to orthogonal polynomials. As offshoots of our analysis, we will construct action/angle variables for the finite Ablowitz‐Ladik hierarchy and describe the long‐time behavior of this system. © 2006 Wiley Periodicals, Inc.