  The Universal T-matrix is the capstone of the structure that consists of a
quantum group and its dual, and the central object from which spring the
T-matrices (monodromies) of all the associated integrable models. A closed
expression is obtained for the case of multiparameter (twisted) quantum
$gl(N)$. The factorized nature of standard quantum groups, that allows the
explicit expression for $U\hskip-1mm T $ to be obtained with relative ease,
extends to some nonstandard quantum groups, such as those based on $A_n^{(2)}$,
and perhaps to all. The paper is mostly concerned with parameters in general
position, but the extension to roots of unity is also explored, in the case of
$g\ell(N)$. The structure of the dual is now radically different, and an
interesting generalization of the $q$-exponential appears in the formulas for
the Universal T- and R-matrices. The projection to quantum $sl(N)$ is simple
and direct; this allows, in particular, to apply recent results concerning
deformations of twisted $gl(N)$ to the semisimple quotient.
