  We introduce a quasitriangular Hopf algebra or `quantum group' $U(B)$, the
{\em double-bosonisation}, associated to every braided group $B$ in the
category of $H$-modules over a quasitriangular Hopf algebra $H$, such that $B$
appears as the `positive root space', $H$ as the `Cartan subalgebra' and the
dual braided group $B^*$ as the `negative root space' of $U(B)$. The choice
$B=f$ recovers Lusztig's construction of $U_q(g)$, where $f$ is Lusztig's
algebra associated to a Cartan datum; other choices give more novel quantum
groups. As an application, our construction provides a canonical way of
building up quantum groups from smaller ones by repeatedly extending their
positive and negative root spaces by linear braided groups; we explicitly
construct $U_q(sl_3)$ from $U_q(sl_2)$ by this method, extending it by the
quantum-braided plane $A_q^2$. We provide a fundamental representation of
$U(B)$ in $B$. A projection from the quantum double, a theory of double
biproducts and a Tannaka-Krein reconstruction point of view are also provided.
