  Unstable particles cannot be treated as asymptotic external states in
$S$-matrix theory and when they occur as resonant states cannot be described by
finite-order perturbation theory. The known facts concerning unstable particles
are reviewed and it is shown how to construct gauge-invariant expressions for
matrix elements containing intermediate unstable particles and physically
meaningful production cross-sections for unstable particles. The results and
methodology presented are relevant for $Z^0$ resonance physics, $W^+W^-$ and
$Z^0Z^0$ pair production and can be straightforwardly applied to other
processes.
