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In this book we examine two kinds of applications in terms of stability and perform numerical evaluations and benchmarks on parallel platforms. We consider the applicability of pseudospectra in the field of hydrodynamic stability to obtain more information than a traditional linear stability analysis can provide. We present parallel computational techniques as well as evaluations of pseudospectra of complex flow processes. In the field of nuclear engineering we treat the criticality problem which describes the crucial criteria for the stability of a nuclear reactor. For the resulting eigenvalue problem we highlight the Davidson method as an attractive alternative to the so far widely used power method.