  Inspired by classical ("actual") Quantum Theory over $\mathbb{C}$ and Modal
Quantum Theory (MQT), which is a model of Quantum Theory over certain finite
fields, we introduce General Quantum Theory as a Quantum Theory -- in the
K{\o}benhavn interpretation -- over general division rings with involution, in
which the inner product "is" a $(\sigma,1)$-Hermitian form $\varphi$. This
unites all known such approaches in one and the same theory, and we show that
many of the known results such as no-cloning, no-deleting, quantum
teleportation and super-dense quantum coding, which are known in classical
Quantum Theory over $\mathbb{C}$ and in some MQTs, hold for any General Quantum
Theory. On the other hand, in many General Quantum Theories, a geometrical
object which we call "quantum kernel" arises, which is invariant under the
unitary group $\mathbf{U}(V,\varphi)$, and which carries the geometry of a
so-called polar space. We use this object to construct new quantum
(teleportation) coding schemes, which mix quantum theory with the geometry of
the quantum kernel (and the action of the unitary group). We also show that in
characteristic $0$, every General Quantum Theory over an algebraically closed
field behaves like classical Quantum Theory over $\mathbb{C}$ at many levels,
and that all such theories share one model, which we pin down as the "minimal
model," which is countable and defined over $\overline{\mathbb{Q}}$. Moreover,
to make the analogy with classical Quantum Theory even more striking, we show
that Born's rule holds in any such theory. So all such theories are not modal
at all. Finally, we obtain an extension theory for General Quantum Theories in
characteristic $0$ which allows one to extend any such theory over
algebraically closed fields (such as classical complex Quantum Theory) to
larger theories in which a quantum kernel is present.
