The most basic structure of chiral conformal field theory (CFT) is the Verlinde ring. This describes the primary fields and mathematically can be represented through a modular tensor category which in practice can be constructed as the representation theory of vertex operator algebras or conformal nets of factors on the circle. Freed-Hopkins-Teleman have expressed the Verlinde ring for the CFTs associated to loop groups as twisted equivariant K-theory. In joint work with Terry Gannon, we build on their work to express K-theoretically the structures of full CFT. In particular, the modular invariant partition functions (which essentially parametrise the possible full CFTs) have a rich interpretation within von Neumann algebras (subfactors), which has led to the developments of structures of full CFT such as the full system (fusion ring of defect lines), nimrep (cylindrical partition function), alpha-induction etc. After an introduction to twisted K-theory, we describe recent work on using bivariant KK-theory of Kasparov to describe models from the doubles of finite groups and lattice models.