  We introduce K\"ahler-Poisson algebras as analogues of algebras of smooth
functions on K\"ahler manifolds, and prove that they share several properties
with their classical counterparts on an algebraic level. For instance, the
module of inner derivations of a K\"ahler-Poisson algebra is a finitely
generated projective module, and allows for a unique metric and torsion-free
connection whose curvature enjoys all the classical symmetries. Moreover,
starting from a large class of Poisson algebras, we show that every algebra has
an associated K\"ahler-Poisson algebra constructed as a localization. At the
end, detailed examples are provided in order to illustrate the novel concepts.
