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A new framework for noncommutative complex geometry on quantum homogeneous spaces is introduced. The main ingredients used are covariant differential calculi and Takeuchi's categorical equivalence for quantum homogeneous spaces. A number of basic results are established, producing a simple set of necessary and sufficient conditions for noncommutative complex structures to exist. Throughout, the framework is applied to the quantum projective spaces endowed with the Heckenberger-Kolb calculus. * Supported by the Grant GACR P201/12/G028 Let A be a unital algebra (in what follows all algebras are assumed to be unital). A first-order differential calculus over A is a pair (Ω 1 , d), where Ω 1 is an A-A-bimodule and d : A → Ω 1 is a linear map for which the Leibniz rule holds d(ab) = a(db) + (da)b, a, b, ∈ A, and for which Ω 1 = span C {adb | a, b ∈ A}. (Where no confusion arises, we will drop explicit reference to d and denote a calculus by its bimodule Ω 1 alone.) We call an

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We realise Heckenberger and Kolb's canonical calculus on quantum projective