Abstract: We begin this series of lectures with an introduction to the general theory of Hopf algebras, their relationship to classical algebraic groups, and the definition of Drinfeld-Jimbo quantised enveloping algebras. The notion of a quantum homogeneous space is then introduced and the classical equivalence between equivariant vector bundles and representations of the isotropy subgroup is shown to carry over to the noncommutative setting. This brings us naturally to the notions of covariant differential calculi, noncommutative complex structures, and noncommutative K\"ahler structures. Throughout, the Podle\'s sphere, or the quantum projective line, is used as a motivating example. Moreover, the natural $q$-deformed Dirac operator associated to the Podle\'s sphere is used to introduce the theory of spectral triples and the $C^*$-algebraic approach to noncommutative geometry. We finish by looking at the Borel--Weil theorem for the Podle\'s sphere and give a brief introduction to the theory of noncommutative projective algebraic geometry.