In this talk I'll discuss how Lagrangian particle methods are being used to study the dynamics of fluid vortices. These methods use the Biot-Savart integral to recover the velocity from the vorticity and they track the flow map using adaptive particle discretizations. I'll present computations of vortex sheet motion in 2D flow, with reference to Kelvin-Helmholtz instability, the Moore singularity, spiral roll-up, and chaotic dynamics. Other examples include vortex rings in 3D flow, and vortex dynamics on a rotating sphere.