  We present a fully flavour-covariant formalism for transport phenomena, by
deriving Markovian master equations that describe the time-evolution of
particle number densities in a statistical ensemble with arbitrary flavour
content. As an application of this general formalism, we study flavour effects
in a scenario of resonant leptogenesis (RL) and obtain the flavour-covariant
evolution equations for heavy-neutrino and lepton number densities. This
provides a complete and unified description of RL, capturing three distinct
physical phenomena: (i) the resonant mixing between the heavy-neutrino states,
(ii) coherent oscillations between different heavy-neutrino flavours, and (iii)
quantum decoherence effects in the charged-lepton sector. To illustrate the
importance of this formalism, we numerically solve the flavour-covariant rate
equations for a minimal RL model and show that the total lepton asymmetry can
be enhanced by up to one order of magnitude, as compared to that obtained from
flavour-diagonal or partially flavour off-diagonal rate equations. Thus, the
viable RL model parameter space is enlarged, thereby enhancing further the
prospects of probing a common origin of neutrino masses and the baryon
asymmetry in the Universe at the LHC, as well as in low-energy experiments
searching for lepton flavour and number violation. The key new ingredients in
our flavour-covariant formalism are rank-4 rate tensors, which are required for
the consistency of our flavour-mixing treatment, as shown by an explicit
calculation of the relevant transition amplitudes by generalizing the optical
theorem. We also provide a geometric and physical interpretation of the
heavy-neutrino degeneracy limits in the minimal RL scenario. Finally, we
comment on the consistency of various suggested forms for the heavy-neutrino
self-energy regulator in the lepton-number conserving limit.
