The QCDSP and QCDOC computers are two generations of multithousand-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 category. 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 six-dimensional mesh and compute nodes fabricated with IBM system-on-a-chip technology. It offers a tenfold improvement in price/performance. 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.