We give a pedagogical introduction to the differential calculus on quantum groups by stressing at all stages the connection with the classical case. We further analyze the relation between differential calculus and quantum Lie algebra of left (right) invariant vectorfields. Equivalent definitions of bicovariant differential calculus are studied and their geometrical interpretation is explained. From these data we construct and analyze the space of vectorfields, and naturally introduce a contraction operator and a Lie derivative, whose properties are discussed.