  In this paper we extend the Falicov-Kimball model (FKM) to the case where the
"quasi-particles" entering the FKM are not "ordinary" fermions. As an example
we first discuss how the FKM can be generalized to the case with spin-dependent
hopping. Afterwards we discuss several cases where the "quasi-particles"
entering the FKM are Majorana fermions (extended Majorana-Falicov-Kimball Model
(MFKM). Two examples of extended MFKM are discussed in detail: (i) a $p$-wave
BCS superconductor on a bipartite lattice and (ii) a BCS-Anderson model. We
also discuss the most general forms of extended MFKM, including a brief
discussion on the case where the Majorana fermions represent spins, but not
real fermion particles.
