We discuss the fictitious domain solution of the Navier-Stokes equations modeling unsteady incompressible viscous flow. The method is based on a Lagrange multiplier treatment of the boundary conditions to be satisfied and is particularly well suited to the treatment of no-slip boundary conditions. This approach allows the use of structured meshes and fast specialized solvers for problems on complicated geometries. Another interesting feature of the fictitious domain approach is that it allows the solution of optimal shape problems without regridding. The resulting methodology is applied to the solution of flow problems including external viscous flow past oscillating rigid body and vortex dynamics of two-dimensional flow modeled by the incompressible Navier-Stokes equations and then to an optimal shape problem for Stokes and Navier-Stokes flows.