Abstract
The Heun–Racah and Heun–Bannai–Ito algebras are introduced. Specializations of these algebras are seen to be realized by the operators obtained by applying the algebraic Heun construct to the bispectral operators of the Racah and Bannai–Ito polynomials. The study supplements the results on the Heun–Askey–Wilson algebra and completes the description of the Heun algebras associated with the polynomial families at the top of the Askey scheme, its q-analog, and the Bannai–Ito one.