the twisted k-theory of a space was introduced by donovan-karoubi (1970) for torsion twistings and by rosenberg (1989) in general. motivated by applications in mathematical physics, the theory has attracted a lot of interest in recent years. in this talk i will review some of the foundations of twisted k-homology, and outline a new proof of the freed-hopkins-teleman theorem describing the twisted equivariant k-homology of simple, simply connected compact lie groups.