Abstract: The geometric invariant theory quotient of a complex affine space by a reductive group can be identified with a symplectic quotient by the associated maximal compact via the Kempf-Ness theorem. The reductive quotient also has a hyperkähler analogue constructed as a hyperkähler quotient. Both the symplectic and hyperkähler quotients are constructed using moment maps for the action. For non-reductive group actions, we ask if the there are also links to suitable symplectic and hyperkähler quotients. We focus our attention on the simplest non-reductive group, the additive group, as many examples of interest involve additive group actions. We start by reviewing the theory for reductive groups and see that for linear additive group actions the same results hold. Then we study non-linear additive group actions that look like a family of linear actions parametrised by a base. In both the symplectic and hyperkähler setting, we provide 'moment maps' in a modified sense. This is joint work in progress with Brent Doran.