  We introduce Heun algebras of Lie type. They are obtained from bispectral
pairs associated to simple or solvable Lie algebras of dimension three or four.
For $\mathfrak{su}(2)$, this leads to the Heun-Krawtchouk algebra. The
corresponding Heun-Krawtchouk operator is identified as the Hamiltonian of the
quantum analogue of the Zhukovski-Voltera gyrostat. For $\mathfrak{su}(1,1)$,
one obtains the Heun algebras attached to the Meixner, Meixner-Pollaczek and
Laguerre polynomials. These Heun algebras are shown to be isomorphic the the
Hahn algebra. Focusing on the harmonic oscillator algebra $\mathfrak{ho}$ leads
to the Heun-Charlier algebra. The connections to orthogonal polynomials are
achieved through realizations of the underlying Lie algebras in terms of
difference and differential operators. In the $\mathfrak{su}(1,1)$ cases, it is
observed that the Heun operator can be transformed into the Hahn, Continuous
Hahn and Confluent Heun operators respectively.
