Particle methods are related to the Green's function approach for solving differential equations and they represent an alternative to the traditional methods of scientific computing such as finite-difference, finite-element, and spectral methods. In this talk we describe some recent developments in particle methods. First we present a treecode algorithm for evaluating multiquadric radial basis function approximations that reduces the operation count from $O(N2)$ to $O(N\log N)$, where $N$ is the number of nodes in the system. The algorithm is based on Cartesian Taylor expansions as opposed to the Laurent expansions used by previous investigators. Second we discuss Lagrangian particle simulations of vortex sheet roll-up in 2D and 3D fluid flow. The Lagrangian approach keeps track of the flow map and we employ special techniques including kernel smoothing for stability, adaptive interpolation for accuracy, and a treecode for efficiency. We consider the Kelvin-Helmholtz problem for a periodic shear layer in 2D and the oblique collision of two vortex rings in 3D.