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N = 2 supersymmetry in four spacetime dimensions is intimately related to hyperkähler and quaternionic Kähler geometries. On one hand, the target spaces for rigid supersymmetric sigma-models are necessarily hyperkähler manifolds. On the other hand, when coupled to N = 2 supergravity, the sigma-model target spaces must be quaternionic Kähler. It is known that such manifolds of restricted holonomy are difficult to generate explicitly. Projective superspace is a field-theoretic approach to construct general N = 2 supersymmetric nonlinear sigma-models, and hence to generate new hyperkähler and quaternionic Kähler metrics. Intended for a mixed audience consisting of both physicists and mathematicians, these lectures provide a pedagogical introduction to the projective-superspace approach.

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