Nonabelian varieties and wild monodromy Every smooth compact complex algebraic curve X has an abelian variety attached to it: its Jacobian. Topologically this can be viewed as the space of representations of the fundamental group of X into the one dimensional unitary group U(1), the circle. This suggests the non-abelian generalisation: replace U(1) by U(n) for n>1, and leads to the Mumford-Narasimhan-Seshadri moduli spaces of stable vector bundles, related to Andre Weil's project of constructing nonabelian theta functions. More recently Hitchin and others looked at a complexified set-up, replacing U(n) by GL_n(C). This leads to a rich class of non-compact non-abelian moduli spaces. I'll explain how this story extends to the case where X is allowed to be non-compact, focusing mainly on the topological and symplectic descriptions of the moduli spaces, and the notion of {\em wild monodromy representations}, generalising the familiar fundamental group representations above.