  We develop a theory of Darboux transformations for CMV matrices, canonical
representations of the unitary operators. In perfect analogy with their
self-adjoint version -- the Darboux transformations of Jacobi matrices -- they
are equivalent to Laurent polynomial modifications of the underlying measures.
We address other questions which emphasize the similarities between Darboux
transformations for Jacobi and CMV matrices, like their (almost) isospectrality
or the relation that they establish between the corresponding orthogonal
polynomials, showing also that both transformations are connected by the
Szeg\"o mapping.
  Nevertheless, we uncover some features of the Darboux transformations for CMV
matrices which are in striking contrast with those of the Jacobi case. In
particular, when applied to CMV matrices, the matrix realization of the inverse
Darboux transformations -- what we call 'Darboux transformations with
parameters' -- leads to spurious solutions whose interpretation deserves future
research. Such spurious solutions are neither unitary nor band matrices, so
Darboux transformations for CMV matrices are much more subject to the
subtleties of the algebra of infinite matrices than their Jacobi counterparts.
  A key role in our theory is played by the Cholesky factorizations of infinite
matrices. Actually, the Darboux transformations introduced in this paper are
based on the Cholesky factorizations of degree one Hermitian Laurent
polynomials evaluated on CMV matrices. We also show how these transformations
can be generalized to higher degree Laurent polynomials, as well as to the
extension of CMV matrices to quasi-definite functionals -- what we call
'quasi-CMV' matrices.
