This abstract focuses on topology optimization for unsteady surface flow of incompressible fluid at low and moderate Reynolds numbers. The unsteady flow is defined on two-dimensional manifolds (2-manifolds). It corresponds to the surface flow. For the volume flow, we have formulated the topology optimization problem by adding the artificial Darcy frictional force into the incompressible Navier-Stokes equations for volume flow by using the material distribution method. The optimization procedure was implemented by using the continuous adjoint method and the mixed finite element method. The effects of dynamic inflow, Reynolds number and target flux on specified boundaries for the optimal topology of unsteady Navier-Stokes flow have been presented for volume flow and its two-dimensional counterpart defined on simplified plane. Being different from volume flow, surface flow represents the motions of the viscous and incompressible fluid at the solid/fluid interfaces. This abstract thereby develops topology optimization for surface flow and extends the design space of topology optimization of fluidic structures onto the curved surfaces in the forms of 2-manifolds. The presented approach is implemented by filling a porous medium onto the 2-manifolds. The artificial Darcy friction is added to the area force term of the surface Navier-Stokes equations used to describe the surface flow. Topology optimization for surface flow is executed by iteratively evolving the impermeability of the porous medium, where the impermeability is interpolated by the material density derived from a design variable. The related partial differential equations are solved by using the surface finite element method. Numerical examples are provided to demonstrate topology optimization for the surface flow, including the boundary velocity driven flow, the area force driven flow and the convection-diffusion flow.