The QCDSP and QCDOC computers are two generations of midtithousand-node multidimensional mesh-based computers designed to study quantum chromodynamics (QCD), the theory of the strong nuclear force. QCDSP (QCD on digital signal processors). a four-dimensional mesh machine, was completed in 1998; in that year, it won the Gordon Bell prize in the price/performance categorly. Two large installations - of 8,192 and 12,288 nodes, with a combined peak speed of one teraflops-have been in operation since. QCD-on-a-chip (QCDOC) utilizes a sixdimensional mesh and compute nodes fabricated with IBM system-on-a-chip technology. It offers a tenfold impropement in price/performace . Currently 100-node versions are operating, and there are plans to build three 12,288-node, 10-teraflops machines. In this paper, we describe the architecture of both the QCDSP and QCDOC machines, the operating systems employed, the user software environment, and the performance of our application-lattice QCD.