In the past few years, a new transistor has appeared on the scene, made of GaAs and AlGaAs, which now holds the record as the fastest logic switching device, switching at speeds of close to ten trillionths of a second (10 ps). The device evolved from the work on GaAs-AlGaAs superlattices (thin alternating layers of differing materials sharing the same crystalline lattice) pioneered by L. Esaki and R. Tsu at IBM in the late 1960's. They realized that high mobilities in GaAs could be achieved if electrons were transferred from the doped and wider band-gap AlGaAs to an adjacent undoped GaAs layer, a process now called modulation doping. R. Dingle, H. L. Stormer, A. C. Gossard, and W. Wiegmann of AT&T Bell Labs, working independently, were the first to demonstrate high mobilities obtained by modulation doping in 1978, in a GaAs-AlGaAs superlattice. Realizing that such a structure could form the basis for a high-performance field-effect transistor (Bell Labs Patent 4163237, filed on April 24, 1978), researchers at various labs in the United States (Bell Labs, University of illinois, and Rockwell), Japan (Fujitsu), and France (Thomson CSF) began working on this device. In 1980, the first such device with a reasonable microwave performance was fabricated by the University of Illinois and Rockwell, which they called a modulation-doped FET or MODFET. The same year Fujitsu reported the results obtained in a device with a 400-µm gate which they called the "high electron mobility transistor" or HEMT, in the open literature. Thomson CSF published shortly thereafter calling their realization a "two-dimensional electron gas FET" or TEGFET, and Bell Labs followed, using the name "selectively doped heterojunction transistor" or SDHT. These names are all descriptive of various aspects of the device operation as we will discuss in the text. For the sake of internal consistency will call it MODFET, hereafter. In this paper we review the principals of MODFET operation, factors affecting its performance, optimization of the device, and comparison with other high-performance compound and elemental semiconductor devices. Finally, the remaining problems and future challenges are pointed out.