  We systematically investigate the non-Hermitian generalisations of the
Landau-Zener (LZ) transition and the Landau-Zener-St\"{u}ckelberg (LZS)
interferometry. The LZ transition probabilities, or band populations, are
calculated for a generic non-Hermitian model and their asymptotic behaviour
analysed. We then focus on non-Hermitian systems with a real adiabatic
parameter and study the LZS interferometry formed out of two identical avoided
level crossings. Four distinctive cases of interferometry are identified and
the analytic formulae for the transition probabilities are calculated for each
case. The differences and similarities between the non-Hermitian case and its
Hermitian counterpart are emphasised. In particular, the geometrical phase
originated from the sign change of the mass term at the two level crossings is
still present in the non-Hermitian system, indicating its robustness against
the non-Hermiticity. We further apply our non-Hermitian LZS theory to
describing the Bloch oscillation in one-dimensional parity-time
$(\mathcal{PT})$ reversal symmetric non-Hermitian Su-Schrieffer-Heeger model
and propose an experimental scheme to simulate such dynamics using photonic
waveguide arrays. The Landau-Zener transition, as well as the LZS
interferometry, can be visualised through the beam intensity profile and the
transition probabilitiess measured by the centre of mass of the profile.
