  Building on the theory of noncommutative complex structures, the notion of a
noncommutative K\"ahler structure is introduced. In the quantum homogeneous
space case many of the fundamental results of classical K\"ahler geometry are
shown to follow from the existence of such a structure, allowing for the
definition of noncommutative Lefschetz, Hodge, K\"ahler-Dirac, and Laplace
operators. Quantum projective space, endowed with its Heckenberger-Kolb
calculus, is taken as the motivating example. The general theory is then used
to show that the calculus has cohomology groups of at least classical
dimension.
