Test and Diagnosis are critical areas in semiconductor manufacturing. Every chip manufactured using a new or premature technology or process needs to be tested for manufacturing defects to ensure defective chips are not sold to the customer. Conventionally, test is done by mounting the chip on an Automated Test Equipment (ATE) and applying test patterns to test for different faults. With shrinking feature sizes, the complexity of the circuits on chip is increasing, which in turn increases the number of test patterns needed to test the chip comprehensively. This increases the test application time which further increases the cost of test, ultimately leading to increase in the cost per device. Furthermore, chips that fail during test need to be diagnosed to determine the cause of the failure so that the manufacturing process can be improved to increase the yield. With increase in the size and complexity of the circuits, diagnosis is becoming an even more challenging and time consuming process. Fast diagnosis of failing chips can help in reducing the ramp-up to the high volume manufacturing stage and thus reduce the time to market. To reduce the time needed for diagnosis, efficient diagnostic patterns have to be generated that can distinguish between several faults. However, in order to reduce the test application time, the total number of patterns should be minimized. We propose a technique for generating diagnostic patterns that are inherently compact. Experimental results show up to 73% reduction in the number of diagnostic patterns needed to distinguish all faults. Logic Built-in Self-Test (LBIST) is an alternative methodology for testing, wherein all components needed to test the chip are on the chip itself. This eliminates the need of expensive ATEs and allows for at-speed testing of chips. However, there is hardware overhead incurred in storing deterministic test patterns on chip and failing chips are hard to diagnose in this LBIST architecture due to limited observability. We propose a technique to reduce the number of patterns needed to be stored on chip and thus reduce the hardware overhead. We also propose a new LBIST architecture which increases the diagnosability in LBIST with a minimal hardware overhead. These two techniques overcome the disadvantages of LBIST and can make LBIST more popular solution for testing of chips. Modern designs may contain a large number of small embedded memories. Memory Built-in Self-Test (MBIST) is the conventional technique of testing memories, but it incurs hardware overhead. Using MBIST for small embedded memories is impractical as the hardware overhead would be significantly high. Test generation for such circuits is difficult because the fault effect needs to be propagated through the memory. We propose a new technique for testing of circuits with embedded memories. By using SMT solver, we model memory at a high level of abstraction using theory of array, while keeping the surrounding logic at gate level. This effectively converts the test generation problem into a combinational test generation problem and make test generation easier than the conventional techniques. / Ph. D.