  We study superconductor-barrier-superconductor (S-B-S) Josephson junctions
constructed out of two-dimensional and three-dimensional triple-point
semimetals, which feature a threefold degeneracy at a single nodal point. We
assume a weak and homogeneous s-wave pairing in each superconducting region,
and a potential difference is applied across a piece of normal-state semimetal
to create the barrier region. We apply the BdG formalism to compute the
wavefunctions of the subgap Andreev bound states (ABSs), considering the
thin-barrier limit, in a piecewise-continuous manner. The appropriate boundary
conditions at the S-B and B-S junctions allow us to compute the discrete energy
eigenvalues $\pm |\varepsilon| $ of the ABSs. We get two distinct solutions for
$|\varepsilon| $. This result is different from the S-B-S set-ups in graphene
and Weyl semimetals (each featuring twofold-degenerate nodal points), where one
obtains only one solution for $|\varepsilon| $. Although the three bands of the
triple-point fermions comprise a nondispersive flat band, the multifold nature
of the bandstructure is responsible for this difference. We also illustrate the
behaviour of the Josephson current flowing across the S-B-S junction,
contributed by the two pairs of ABSs.
