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.