  We present the Usadel theory describing the superconducting proximity effect
in heterostructures with a half-metallic layer. It is shown that the full spin
polarization inside the half-metals gives rise to the giant triplet spin-valve
effect in superconductor (S) - ferromagnet (F) - half-metal (HM) trilayers as
well as to the $\varphi_0$-junction formation in the S/F/HM/F/S systems. In
addition, we consider the exactly solvable model of the S/F/HM trilayers of
atomic thickness and demonstrate that it reproduces the main features of the
spin-valve effect found within the Usadel approach. Our results are shown to be
in a qualitative agreement with the recent experimental data on the spin-valve
effect in ${\rm MoGe/Cu/Ni/CrO_2}$ hybrids [A. Singh et al., Phys. Rev. X 5,
021019 (2015)].
