  We review the notion of complementarity of observables in quantum mechanics,
as formulated and studied by Paul Busch and his colleagues over the years. In
addition, we provide further clarification on the operational meaning of the
concept, and present several characterisations of complementarity - some of
which new - in a unified manner, as a consequence of a basic factorisation
lemma for quantum effects. We work out several applications, including the
canonical cases of position-momentum, position-energy, number-phase, as well as
periodic observables relevant to spatial interferometry. We close the paper
with some considerations of complementarity in a noisy setting, focusing
especially on the case of convolutions of position and momentum, which was a
recurring topic in Paul's work on operational formulation of quantum
measurements and central to his philosophy of unsharp reality.
