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In the presence of classical phase space singularities the standard methodsare insufficient to attack the problem of quantization.In certain situationsthe difficulties can be overcome by means of K\"ahler quantization onstratified K\"ahler spaces. Such a space is a stratified symplectic spacetogether with a complex analytic structure which is compatible with thestratified symplectic structure; in particular each stratum is a K\"ahlermanifold in an obvious fashion. Holomorphic quantization on a stratifiedK\"ahler space then yields a costratified Hilbert space, a quantum objecthaving the classical singularities as its shadow. Given a K\"ahler manifoldwith a hamiltonian action of a compact Lie group that also preserves thecomplex structure, reduction after quantization coincides with quantizationafter reduction in the sense that not only the reduced and unreduced quantumphase spaces correspond but the invariant unreduced and reduced quantumobservables as well

Singular Poisson-K\"ahler geometry of stratified K\"ahler spaces and quantization