Many interactive rendering algorithms require operations on multiple fragments (i.e., ray intersections) at the same pixel location: however, current Graphics Processing Units (GPUs) capture only a single fragment per pixel. Example effects include transparency, translucency, constructive solid geometry, depth-of-field, direct volume rendering, and isosurface visualization. With current GPUs, programmers implement these effects using multiple passes over the scene geometry, often substantially limiting performance. This paper introduces a generalization of the Z-buffer, called the k-buffer, that makes it possible to efficiently implement such algorithms with only a single geometry pass, yet requires only a small, fixed amount of additional memory. The k-buffer uses framebuffer memory as a read-modify-write (RMW) pool of k entries whose use is programmatically defined by a small k-buffer program. We present two proposals for adding k-buffer support to future GPUs and demonstrate numerous multiple-fragment, single-pass graphics algorithms running on both a software-simulated k-buffer and a k-buffer implemented with current GPUs. The goal of this work is to demonstrate the large number of graphics algorithms that the k-buffer enables and that the efficiency is superior to current multipass approaches.