  We give a pedagogical introduction to the differential calculus on quantum
groups by stressing at all stages the connection with the classical case ($q
\rightarrow 1$ limit). The Lie derivative and the contraction operator on forms
and tensor fields are found. A new, explicit form of the Cartan--Maurer
equations is presented. The example of a bicovariant differential calculus on
the quantum group $GL_q(2)$ is given in detail. The softening of a quantum
group is considered, and we introduce $q$-curvatures satisfying q-Bianchi
identities, a basic ingredient for the construction of $q$-gravity and
$q$-gauge theories.
