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We present novel and efficient methods for built-in self-test (BIST) of field-programmable gate arrays (FPGAs) for detection and diagnosis of permanent faults in current, as well as emerging, technologies that are expected to have high fault densities. Our basic BIST methods can be used in both online and offline testing scenarios, although we focus on the former in this paper. We present 1- and 2-diagnosable BISTer designs that make up a ROving TEster (ROTE). Due to their provable diagnosabilities, these BISTers can avoid time-intensive adaptive diagnosis without significantly compromising diagnostic coverage-the percentage of faults correctly diagnosed. We also develop functional testing methods that test programmable logic blocks (PLBs) in only two circuit functions that will be mapped to them as the ROTE moves across a functioning FPGA. We extend our basic BISTer designs to those with test-pattern generators (TPGs) using multiple PLBs to more efficiently test the complex PLBs of current commercial FPGAs and to also prove the diagnosabilities of these designs. Simulation results show that our 1-diagnosable functional-test-based BISTer with a three-PLB TPG has very high diagnostic coverages-for example, for a random-fault distribution, our nonadaptive-diagnosis methods provide diagnostic coverages of 96% and 88% at fault densities of 10% and 25%, respectively, whereas the previous best nonadaptive-diagnosis method of the STAR-3 × 2 BISTer has diagnostic coverages of about 75% and 55% at these fault densities.