This talk describes recent Lagrangian simulations of incompressible fluids and collisionless electrostatic plasmas. In both cases, the standard Eulerian formulation is replaced by a Lagrangian formulation in terms of the flow map. This leads naturally to a particle discretization. The particles carry vorticity in the case of a fluid and electric charge in the case of a plasma. The induced velocity and electric field are expressed as singular integrals. The numerical method uses kernel regularization for stability, adaptive particle insertion for accuracy, and a multipole treecode for efficiency. Examples to be presented include electron beams in 1D plasmas, and vortex sheets and vortex rings in 2D and 3D fluids. The Lagrangian approach gives direct access to dynamics, revealing the onset of chaos in these flows.