With rapid advancement in science and technology and decreasing feature size of transistors, the complexity of VLSI designs is constantly increasing. With increasing density and complexity of the designs, the probability of occurrence of defects also increases. Therefore testing of designs becomes essential in order to guarantee fault-free operation of devices. Testing of VLSI designs involves generation of test patterns, test pattern application and identification of defects in design. In case of scan based designs, the test set size directly impacts the test application time which is determined by the number of memory elements in the design and the test storage requirements. There are various methods in literature which are used to address the issue of large test set size classified as static or dynamic compaction methods depending on whether the test compaction algorithm is performed as a post-processing step after test generation or is integrated within the test generation. In general, there is a trade-off between the test compaction achievable and the run-time. Methods which are computationally intensive might provide better compaction, however, might have longer run times owing to the complexity of the algorithm. In the first part of the thesis we address the problem of large test set size in partially scanned designs by proposing an incremental dynamic compaction method. Typically, the fault coverage curve of designs ramp up very quickly in the beginning and later slows down and ultimately the curve flattens towards the tail of the curve. In the initial phase of test generation a greedy compaction method is used because initially there are easy-to-detect faults and the scope for compaction is better. However, in the later portion of the curve, there are hard-to-detect faults which affect compaction and we propose to use a dynamic compaction approach. We propose a novel mechanism to identify redundant faults during dynamic compaction to avoid targeting them later. The effectiveness of method is demonstrated on industrial designs and test size reduction of 30% is achieved. As the device complexity is increasing, delay defects are also increasing. Speed path debug is necessary in order to meet performance requirements. Speed paths are the frequency limiting paths in a design identified during debug. Speed paths can be tested using functional patterns, transition n-detect patterns or path delay patterns. However, usage of functional patterns for speed path debug is expensive because generation of functional patterns is expensive and the application cost is also high because the number of patterns is large and requires functional testers. In the second part of the dissertation we propose a simple path sensitization approach that can be used to generate pseudo-robust tests, which are near robust tests and can be used for designs that have multiple clock domains. The fault coverage for path delay fault APTG can be further improved by dividing the paths that are not testable under pseudo robust conditions, into shorter sub-paths. The effectiveness of the method is demonstrated on industrial designs.