  We define some new algebraic structures, termed coloured Hopf algebras, by
combining the coalgebra structures and antipodes of a standard Hopf algebra set
$\cal H$, corresponding to some parameter set $\cal Q$, with the
transformations of an algebra isomorphism group $\cal G$, herein called colour
group. Such transformations are labelled by some colour parameters, taking
values in a colour set $\cal C$. We show that various classes of Hopf algebras,
such as almost cocommutative, coboundary, quasitriangular, and triangular ones,
can be extended into corresponding coloured algebraic structures, and that
coloured quasitriangular Hopf algebras, in particular, are characterized by the
existence of a coloured universal $\cal R$-matrix, satisfying the coloured
Yang-Baxter equation. The present definitions extend those previously
introduced by Ohtsuki, which correspond to some substructures in those cases
where the colour group is abelian. We apply the new concepts to construct
coloured quantum universal enveloping algebras of both semisimple and
nonsemisimple Lie algebras, considering several examples with fixed or varying
parameters. As a by-product, some of the matrix representations of coloured
universal $\cal R$-matrices, derived in the present paper, provide new
solutions of the coloured Yang-Baxter equation, which might be of interest in
the context of integrable models.
