From the point of view of Riemannian geometry, hyperkahler manifolds are manifolds with metrics whose holonomy is contained in the quaternionic unitary group Sp(n) = SO(4n) ∩ GL(n,H), where GL(n,H) is the group of n×n invertible matrices with quaternionic entries. In this talk I will discuss the quaternionic structure and Kahler structures on a hyperkahler manifold, and describe the two main methods of constructing examples: twistor spaces and the quotient construction. I may also also describe how hyperkahler manifolds can be studied from the point of view of complex algebraic geometry.

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