We develop a boundary integral method for computing the motion of an interface separating two incompressible, inviscid fluids. The velocity integral is regularized, so that the vortex sheet on the interface is replaced by a sum of "blobs" of vorticity. The regularization allows control of physical instabilities. We design a class of high order blob methods and analyze the errors. Numerical tests suggest that the blob size should be scaled with the local spacing of the interfacial markers. For a vortex sheet in one fluid, with a first-order kernel, we obtain a spiral roll-up similar to Krasny [J. Comput. Phys. 65 (1986) 292], but the higher order kernels lead to more detailed structure. We verify the accuracy of the new method by computing a liquid-gas interface with Rayleigh-Taylor instability. We then apply the method to the more difficult case of Rayleigh-Taylor flow separating two fluids of positive density, a case for which the regularization appears to be essential, as found by Kerr and Tryggvason [both J. Comput. Phys. 76 (1988) 48; 75 (1988) 253]. We use a "blob" regularization in certain local terms in the evolution equations as well as in the velocity integral. We find strong evidence that improved spatial resolution with fixed blob size leads to a converged, regularized solution without numerical instabilities. However, it is not clear that there is a weak limit as the regularization is decreased. © 2003 Elsevier Inc. All rights reserved.